The Petersen-Estimator (and much more) for Two-Sample Capture-Recapture Studies with Applications to Fisheries Management

Table of contents

• 1 Installation & Change log
  • 1.1 Installation
  • 1.2 Access to accompanying workbook
  • 1.3 Change log
• 2 Introduction
• 3 Basic Sampling protocols and estimation
  • 3.1 Conceptual basis
  • 3.2 Sampling protocols
  • 3.3 Estimation
  • 3.4 Software
  • 3.5 Example - Estimating the number of fish in Rödli Tarn
• 4 Planning studies
• 5 Assumptions and effects of violations
  • 5.1 Introduction
  • 5.2 Non-closure
  • 5.3 Marking has no effect
  • 5.4 Tag loss
  • 5.5 Tags not reported or overlooked
  • 5.6 Homogeneity in catchablity
  • 5.7 Spreadsheet to investigate impact of assumption violations
  • 5.8 Empirical adjustments to existing estimates
• 6 Accounting for heterogeneity I - fixed discrete strata
  • 6.1 Fixed individual categorical covariates such as sex
  • 6.2 Example: Northern Pike - simple stratification
  • 6.3 Example: Northern pike stratified by length class
• 7 Accounting for heterogeneity II - continuous fixed covariate
  • 7.1 Quadratic relationship between the probability of capture and length - Northern Pike
  • 7.2 Spline relationship between the probability of capture and length - Northern Pike
  • 7.3 Non-parametric smoothing
• 8 Accounting for heterogeneity III - geographic or temporal stratification
  • 8.1 Introduction
  • 8.2 Geographic stratification
  • 8.3 Temporal stratification
• 9 Incomplete or partial stratification
  • 9.1 Introduction
  • 9.2 Sampling protocol
  • 9.3 Capture histories and parameters
  • 9.4 Example - Walleye in Milles Lac, MN.
  • 9.5 Optimal allocation
  • 9.6 Additional details
• 10 Double tagging studies to account for tag loss or non-reporting
  • 10.1 Introduction
  • 10.2 Capture histories
  • 10.3 Example - Kookanee on Metolius River - two tags that cannot be distinguished
  • 10.4 Example - Simulated data - two tags that can be distinguished
  • 10.5 Example - Simulated data - a permanent second tag used
  • 10.6 Example - Northern Pike - tag loss
  • 10.7 Example - Simulated data - non-reporting of tags and reward tags
  • 10.8 Planning tag loss studies
• 11 Multiple marks
  • 11.1 Introduction
  • 11.2 Theory
  • 11.3 Illustration of using multimark with simulated data.
  • 11.4 Bobcat example
• 12 Multiple Petersen estimates
  • 12.1 Introduction
  • 12.2 Batch marks (logit-normal model)
  • 12.3 Example of PNE method - TBA
  • 12.4 Summary
• 13 Forward and reverse capture-recapture studies
  • 13.1 Introduction
  • 13.2 Example - Run size of Yukon River Chinook
  • 13.3 Example: Lower Fraser Coho - SPAS in reverse
  • 13.4 Example combining forward and reverse capture-recapture studies
  • 13.5 Summary
• 14 Genetic capture-recapture methods using close kin studies
  • 14.1 Transgenerational capture-recapture
• 15 Grand summary
• 16 Miscellaneous notes
  • 16.1 Speeding computation times
• 17 References
• 18 Appendix A - The multinomial models for the Petersen study
  • 18.1 Complete multinomial model
  • 18.2 Conditional multinomial model
• 19 Appendix B - Using the VGAM package
  • 19.1 Data structure
  • 19.2 Example Rodli Tarn
  • 19.3 Example Rodli Tarn - Chapman correction
  • 19.4 Example - Rodli Tarn - Equal capture probabilities
  • 19.5 Example - Northern Pike - Stratification by Sex
  • 19.6 Example - Northern Pike - Stratification by Length Class
• 20 Appendix C. Using RMark/MARK
  • 20.1 Data structure
  • 20.2 Example Rodli Tarn
  • 20.3 Example Rodli Tarn - Chapman correction
  • 20.4 Example - Rodli Tarn - Equal capture probabilities
  • 20.5 Example - Northern Pike - Stratification by Sex
  • 20.6 Example - Northern Pike - Model with length as a covariate
• 21 Appendix D. Conditional likelihood approach to tag-loss studies.
• 22 Appendix E: Brief theory for BTSPAS models.
  • 22.1 Diagonal case
  • 22.2 Non-diagonal case

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1 Installation & Change log

1.1 Installation

The Petersen package can be installed from CRAN in the usual way.

The development version can be installed from within R using:

devtools::install_github("cschwarz-stat-sfu-ca/Petersen", 
                        dependencies = TRUE,
                        build_vignettes = TRUE)

1.2 Access to accompanying workbook

The accompanying workbook is available in the external data directory of the package and can copied to your working directory using:

extdata.dir <- system.file("extdata", package="Petersen")
extdata.dir

[1] "/Users/cschwarz/Rlibs/Petersen/extdata"

extdata.list <- list.files(extdata.dir)
extdata.list

[1] "PetersenWorkBook-2023-05-01.xls"

# Uncomment the following to copy the PetersenWorkbook to your working directory.
#file.copy(from=file.path(extdata.dir, "PetersenWorkbook-2006-01-20.xls"), to=getwd())

1.3 Change log

Date	Changes
2025-03-01	Added n1_n2_m2_to_cap_hist() function to generate capture histories summary statistics
2024-06-01	Added new example of forward-reverse capture-recapture (lower Fraser coho)
	Added chapter on use of multimark when you have left and right photographs and cannot match left and right to the same animal.
2023-12-01	Updates for CRAN submission. No new functionality or examples.
2023-05-01	First Release

2 Introduction

In the late 1890’s, Petersen (1896) estimated exploitation probabilities and the abundance of fish living in enclosed bodies of water. His methods have been widely adopted and the Petersen-method is among the most widely used methods in fisheries management.

Similar methods were used by Laplace (1793) to estimate the population of France based on birth registries; by Lincoln (1930) to estimate the abundance of ducks from band returns; and by Jackson (1933) to estimate the density of tsetse flies.

The Petersen-method is the simplest of more general capture-recapture methods which are extensively reviewed in Williams et al. (2002). Despite the Petersen method’s simplicity, many of the properties of the estimator, and the effects of violations of assumptions are similar to these more complex capture-recapture studies. Consequently, a firm understanding of the basic principles learned from studying this method are extremely useful to develop an intuitive understanding of the larger class of studies.

The purpose of this monograph is to bring together a wide body of older and newer literature on the design and analysis of the “simple” two-sample capture-recapture study. This monograph builds upon the comprehensive summaries found in Ricker (1975), Seber (1982), Seber and Schofield (2023), and William et al (2002), and incorporates newer works that have not yet summarized. While the primary emphasis is on the application to fisheries management, the methods are directly applicable to many other situations.

Computer software has revolutionized many aspects of data analysis. A workbook accompanies this monograph to assist in the design and analysis of the Petersen studies. As well, the Petersen package in R is available for download.

Following Stigler’s law of eponymy https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy, Goudie and Goudie (2007) investigated the origin of the Petersen estimator and identified additional scientists whose early work on marking and estimating fish populations deserves more credit than it has received.

3 Basic Sampling protocols and estimation

3.1 Conceptual basis

The fundamental goal of a Petersen study is to estimate \(N\), the number of animals (the abundance) in the population of interest.

The idealized protocol consists of an initial capture of \(n_1\) animals from the population. These animals are given a mark or tag (Figure 1).

Figure 1: Common tag types and tagging locations. Diagram taken from XXXX.

Marks or tags fall into two general categories. Batch marks are simple marks that only identify that an animal was captured at a particular occasion. It is impossible to identify individual animals from batch marks. While batch marking is sufficient for the Petersen and other two-sample methods, modern practice is to use individually numbered tags so that the capture history of each individual animal can be determined and individual animals can be identified from each other.

The marked/tagged animals are then returned to the population. After allowing the marked and remaining unmarked animals to mix, a second sample of size \(n_2\) is selected. Each animal in the second sample is examined, and a count of the number of animals marked in the first sample and now recaptured, \(m_2\) is taken.

The summary statistics \(n_1\), \(n_2\), and \(m_2\) are sufficient statistics for this study. Modern practice is to record information in terms of capture histories rather than these summary statistics. A capture history, \(\omega\) is a vector where component \(i\) of the vector takes the value of \(1\) if an animal was captured at sampling event (time) \(i\) and the value \(0\) if the animal was not captured at the sampling event (time) \(i\). Because the Petersen study only has two capture occasions, all capture histories are of length 2 where:

• \(\omega=\{1,1\}\) represents an animal captured at both sampling occasions.
• \(\omega=\{1,0\}\) represents an animal captured at the first occasion but not at the second occasion.
• \(\omega=\{0,1\}\) represents an animal not captured at the first occasion but captured at the second occasion.
• \(\omega=\{0,0\}\) represents animals not captured at either sampling occasion (not observable).

The notation \(n_\omega\) represents the number of animals with capture history \(\omega\). Note that

• \(N=n_{\{0,0\}}+n_{\{0,1\}}+n_{\{1,0\}}+n_{\{1,1\}}\),
• \(n_1=n_{\{1,0\}}+n_{\{1,1\}}\),
• \(n_2=n_{\{0,1\}}+n_{\{1,1\}}\) and
• \(m_2=n_{\{1,1\}}\).

The capture history notation is especially convenient for more complex capture-recapture studies. Both notations will be used in this monograph.

The fundamental estimating equation is based on the idea that the proportion of marked animals in the second sample should be approximately equal to the proportion of animals initially captured:

\[\frac{m_2}{n_2} \approx \frac{n_1}{N}\]

By rearranging this relation, the Petersen estimator (Equation 1) is obtained

\[\widehat{N}_{Petersen} = \frac{n_1 n_2}{m_2} \tag{1}\]

The estimated capture probabilities at each sample occasion can also be obtained: \[\widehat{p}_1 = \frac{n_1}{\widehat{N}} = \frac{m_2}{n_2}\] \[\widehat{p}_2 = \frac{n_2}{\widehat{N}} = \frac{m_2}{n_1}\]

If \(\widehat{N}\) is to be a sensible estimator of \(N\), several assumptions must be made. The effects of violations of these assumptions will be discussed in Section 5.

Following Seber (1982) these assumptions are:

1. Closure. In other words, no animals leave the population (by death or emigration) and no animals enter the population (by internal births or immigration) between sampling occasions.

2. Homogeneity. This assumption can be satisfied in a number of ways. All animals have the same probability of capture at the first occasion, or all animals have the same probability of capture at the second occasion, or the second sample is a simple random sample selected from the entire population, i.e., each animals is captured or not captured independently of every other animal with the same probability. This assumption allows for some latitude in the study. For example, fish may be conveniently marked in the first sample without worrying about randomization if the scientist is sure that the second sample is a random sample from the entire population. Of course, it is likely wise to make such a strong assumption and randomization in each sample is highly recommended.

3. No impact of marking. Marking the animal does not affects its subsequent survival or catchability, i.e., all animals in the second sample, regardless of marking status, have the same probability of capture.

4. No tag loss. No marked animal loses its mark between the two sampling occasions.

5. No missed tags. All marked animals are correctly identified in the second sample.

3.2 Sampling protocols

While the estimator \(\widehat{N}\) in Equation 1 is intuitively appealing, its properties (bias and precision) cannot be investigated unless the sampling scheme that results in the observed data is fully specified. There are many sampling protocols, the most common being:

• Direct Sampling. In direct sampling method the sample sizes (\(n_1\) and \(n_2\)) are specified in advance.}
• Inverse sampling. In Inverse Sampling, sampling continues until a certain number of marks is obtained.

For each of these protocols, fish can be sampled with or without replacement giving:

• Hypergeometric Sampling. In this methods sampling occurs without replacement;
• Binomial sampling. In the binomial model (sampling with replacement); and

Modern statistical methods for capture-recapture data consider the number of animals in each capture history to be a realization of a multinomial distribution, which is a generalization of the binomial sampling methods.

3.3 Estimation

3.3.1 Maximum likelihood estimation

Once the sampling model has been established, the standard method of obtaining the estimator is Maximum Likelihood Estimation (MLE). The likelihood equations for the various sampling protocols is shown in Table 1.

Maximum Likelihood Estimators asymptotically unbiased and fully efficient (i.e., make use of all of the data and results in the smallest possible standard error). It turns out that for all the sampling schemes discussed above that the estimator \(\widehat{N}\) from Equation 1 is the MLE.

While the MLE is optimal in large samples, it can be severely biased with smaller sample sizes because of the presence of \(m_2\) in the denominator. Indeed, when \(E[m_2]\) is small, there is a fairly large probability that no marks would be observed (i.e., \(m_2=0\)) and \(\widehat{N}=\infty\). Chapman (1951) suggested a simple modification to the estimator as shown in Table 1. Add reference here to Rivest’s work on determining the optimal correction factor.

These modifications removes most of the bias, and Robson and Regier (1964) showed that the approximate residual bias (when \(n_1+n_2 < N\)) is \[b=E\left[\frac{\widehat{N}_{HU}-N}{N}\right] = E\left[- \exp \left( - \frac{(n_1+1)(n_2+1)}{N}\right)\right]\] If \(E[m_2]>4\), the residual relative bias is less than 2% of the abundance. To allow for variation in the observed \(m_2\) around its expected value, Robson and Regier (1964) recommend that studies have \(m_2>7\) to be 95% confident that the \(E[m_2]>4\) and that the residual relative bias in \(\widehat{N}_{HU}\) is negligible. This is equivalent to ensuring that \(n_1 n_2 > 4N\) as outlined by Robson and Regier (1964).

Table 1: Summary of sampling models, bias adjusted estimators, and estimated variance of estimators

Sampling Model	Likelihood	Bias adjusted estimator	Estimated variance
Direct Hypergeometric	\(\frac{\binom{n_1}{m_2} \binom{N-n_1}{n_2-m_2}}{\binom{N}{n_2}}\)	\(\widehat{N} _{HU}=\frac{(n_1+1)(n_2+1)}{(m_2+1)}-1\)	\(\widehat{v}_{HU} = \frac{(n_1+1)(n_2+1)(n_1-m_2)(n_2-m_2)}{(m_2+1)^2(m_2+2)}\)
Direct Binomial	\(\binom{n_2}{m_2}\left(\frac{n_1}{N}\right)^{m_2}\left( 1-\frac{n_1}{N}\right)^{n_2-m_2}\)	\(\widehat{N}_{BU} = \frac{n_1(n_2+1)}{m_2+1}\)	\(\widehat{v}_{BU} = \frac{n_1^2 (n_2+1) (n_2-m_2)}{(m_2+1)^2 (m_2+2)}\)
Inverse Hypergeometric	\(\frac{\binom{n_1}{m_2-1}\binom{N-n_1}{n_2-m_2}}{\binom{N}{n_2-1}} \times \frac{n_1-m_2+1}{N-n_2+1}\)	\(\widehat{N}_{IHU} = \frac{(n_1+1)n_2}{m_2}-1\)	to be added later
Inverse Binomial	\(\binom{n_2-1}{m_2-1} \left(\frac{n_1}{N}\right)^{m_2}\left( 1-\frac{n_1}{N}\right)^{n_2-m_2}\)	\(\widehat{N}_{IBU} = \frac{n_1 n_2}{m_2}\)	\(\widehat{v}_{IBU} = \frac{n_1^2 n_2 (n_2-m_2)}{m_2^2(m_2+1)}\)
Capture History Multinomial	\(\binom{N}{n_{00},n_{01},n_{10},n_{11}} \times\) \(\left( (1-p_1)(1-p_2) \right)^{n_{00}} \times\) \(\left( p_1(1-p_2) \right)^{n_{10}} \times\) \(\left( (1-p_1)p_2 \right)^{n_{01}} \times\) \(\left( p_1 p_2 \right)^{n_{11}}\)	\(\widehat{N}_{MN} = \frac{{\left( {n_{11} + n_{10} } \right)\left( {n_{11} + n_{01} } \right)}}{{n_{11} }} =\) \(\frac{{n_1 n_2 }}{{m_2 }}\)	\(\widehat{v}_{MN} = \frac{{n_1 n_2 }}{{m_2 }}\frac{{\left( {n_2 - m_2 } \right)}}{{m_2 }}\frac{{\left( {n_1 - m_2 } \right)}}{{m_2 }}\)

Chapman (1951) also derived the variance of the hypergeometric estimator and expressed it as: \[V \left( \widehat{N}_{HU} \right) = N^2 \left( E[m_2]^{-1} + 2E[m_2]^{-2} + 6E[m_2]^{-3} \right)\] If \(E[m_2]\) is small, then the variance is large. For example if \(E[m_2]=10\), then the relative standard error (\(RSE=\frac{SE}{estimate})\)) is over 35% and a 95% confidence interval will only be accurate to within 70% of the true value of \(N\)! A discussion of sample size requirements is presented in Section 4.

In cases where the marked fraction is very small (i.e., \(n_1 << N\)) or when sampling is done with replacement (e.g. when animals are observed rather than physically captured), the binomial model of Bailey (1951, 1952) may be used as summarized in Table 1. Seber (1982, p. 61) also points out that the Bailey estimator may be appropriate when complete mixing of marked and unmarked animals takes places and a systematic sample rather than a simple random sample is taken. The Bailey adjusted estimator has comparable residual bias to the Chapman adjusted estimator and is not reported here. In most practical situations, there is little difference between the hypergeometric and binomial sampling model estimates or estimated variances.

In inverse sampling, the number of recaptures (\(m_2\)) to be obtained is fixed in advance and the size of the second sample (\(n_2\)) is now random. This was considered by Bailey (1951), Chapman (1951), Robson (1979); summarized by Seber (1982, Section 3.8); and results are presented in Table 1. Both inverse sampling methods are more efficient (i.e., for a given precision of the estimated abundance, the expected sample size required to obtain this precision under inverse sampling is smaller than that required under direct sampling) than their direct sampling counterparts, but the gain in efficiency is, in practice, negligible. The primary disadvantage of inverse sampling is that with poor planning, the expected sample size could be very large. Seber (1982) summarizes an alternate sampling scheme in which the number of unmarked individuals captured in the second sample is fixed.

Seber (1982) discusses the case of random sample sizes and notes that in practice one would condition upon the observed values so that these previous models are still appropriate. One can also think of a model where the number of fish in the four possible tag histories is random which naturally evolves into more complex studies with multi-samples from both open and closed populations.

In the multinomial model, the counts in the four possible capture histories is considered to arrive from a multinomial model. This multinomial model is the predominant paradigm in current capture-recapture methodology and will be used in the remainder of this monograph. The derivation of the results under the multinomial sampling protocol is found in Section 18.1 and the results are also summarized in Table 1.

In some cases, despite best efforts, no recapture are observed, i.e., \(m_2=0\). While the MLE is nonsensical, the adjusted estimators of Table 1 do provide “estimates” but these will be of very poor precision. Alternatively, Bell (1974) showed that under the hypergeometric sampling model \[P(m_2=0) = \frac{(N-n_1)! (N-m_2)!}{N! (N-n_1-n_2)!}\] and suggested solving \(P(m_2=0)=.5\) as an estimator of \(N\) with the solutions to \(P(m_2=0)=.025\) and \(P(m_2=0)=.975\) as 95% confidence bounds for \(N\). While this works in theory, the resulting estimator has such poor precision that the practical application of this results is doubtful.

Finally, it should be noted, that most sampling plan do not fall exactly into any of the above categories. In many practical plans, the amount of effort (e.g. person-days) is specified for both the initial and second sample, and the number of fish that are captured depends on chance events within that period of effort and should be considered random.

In most studies, results from any of the estimators are similar enough that the practitioner should not be too concerned about the choice of appropriate model – error in the estimates attributable to other problems in the experiment such as assumptions not being satisfied exactly will likely be an order of magnitude greater than any differences in estimates of abundance or precision among the model formulations.

3.3.2 Conditional likelihood estimation

While the MLE are fully efficient, an alternate estimator can be derived that conditions on the observed fish only. Huggins (1989) provides the theory for the conditional multinomial which uses only the observed fish to estimate catchabilities and abundance is estimated using a Horvitz-Thompson type estimator (Section 18.2)

Its main usage is when catchability depends upon individual covariates (e.g., fish length) which is unknown for fish never caught.

In the case of no individual covariates, the conditional likelihood estimators reduce to the full likelihood estimators with no loss of efficiency (i.e., same standard errors), so in practice, the conditional likelihood approach can always be used and is the basis for estimators found in the Petersen package. See Section 18.2 for more details.

3.3.3 Confidence intervals

Confidence intervals for the abundance can be computed in a number of ways.

3.3.3.1 Large sample Wald interval

This method relies upon the asymptotic normality of \(\widehat{N}\) and the usual large-sample, asymptotic, normal-theory based confidence interval is: \[\widehat{N} \pm z_{\alpha/2} SE_{\widehat{N}}\]

However, this interval is not recommended for standard usage for two reasons:

• The distribution of \(\widehat{N}\) is typically skewed with a long right tail;
• There is a very strong positive correlation between \(\widehat{N}\) and the estimated \(SE\). This implies that when \(\widehat{N}\) is below the true population value, the \(SE\) tends to also be smaller than the true \(SE\) and the resulting confidence interval is too narrow, and vice verso.

Many authors have suggested modifications to improve upon the large-sample asymptotic result above.

3.3.3.2 Logarithmic transform

Programs such as MARK compute a confidence interval on the logarithm of \(\widehat{N}\) and then invert the corresponding interval.

\[ \widehat{\theta} = \log{\widehat{N}}\] \[ se_{\widehat{\theta}} = \frac{se_{\widehat{N}}}{\widehat{N}}\] The confidence interval for \(\log{\widehat{N}}\) is found as \[\widehat{\theta} \pm z_{\alpha/2} se_{\widehat{\theta}}\] and then the confidence interval for \(\widehat{N}\) is found by re-inverting the above interval \[\exp{(\widehat{\theta} - z_{\alpha/2} se_{\widehat{\theta}})}~ to~ \exp{(\widehat{\theta} + z_{\alpha/2} se_{\widehat{\theta}})}\]

3.3.3.3 Inverse transformation

Ricker (1975) suggest first using a simple inverse transform, then finding a confidence interval based upon \(1/\widehat{N}\), and then re-inverting the resulting confidence interval, i.e., first compute:

\[ \widehat{\theta} = \frac{1}{\widehat{N}}\] \[ se_{\widehat{\theta}} = \frac{se_{\widehat{N}}}{\widehat{N}^2}\] Then find a confidence interval for \(1/N\) based upon the transformed values, \[\widehat{\theta} \pm z_{\alpha/2} se_{\widehat{\theta}}\] and finally then the confidence interval for \(\widehat{N}\) is found by re-inverting the above interval \[\frac{1}{\widehat{\theta} + z_{\alpha/2} se_{\widehat{\theta}}}~ to ~ \frac{1}{\widehat{\theta} - z_{\alpha/2} se_{\widehat{\theta}}}\]

3.3.3.4 Inverse cube-root transform

The inverse approximation above appears to work fairly well, but Sprott (1981) used likelihood theory to show that the inverse-cube-root transform more effectively captured the skewness in the sampling distribution. His procedure is very similar to the above: \[ \widehat{\theta} = \frac{1}{\widehat{N}^{\frac{1}{3}}}= \frac{1}{\sqrt[3]{\widehat{N}}}\] \[ se_{\widehat{\theta}} = \frac{se_{\widehat{N}}}{3 \widehat{N}^{4/3}}\] The confidence interval for \(1/N^{1/3}\) is found as \[\widehat{\theta} \pm z_{\alpha/2} se_{\widehat{\theta}}\] and then then the confidence interval for \(\widehat{N}\) is found by re-inverting the above interval \[\frac{1}{(\widehat{\theta} + z_{\alpha/2} se_{\widehat{\theta}})^3}~ to \frac{1}{(\widehat{\theta} - z_{\alpha/2} se_{\widehat{\theta}})^3}\]

3.3.3.5 Bootstrapping

Bootstrapping is easy to implement when data is collected as capture-histories (rather than the summary statistics) because each fish is represented by an individual capture history. A resampling of the capture-histories then represents a bootstrap sample; estimates can be computed for each bootstrap sample, and the usually methods for determining confidence intervals from bootstrap samples can be used.

3.3.3.6 Bayesian credible intervals

Add more details later

3.3.3.7 Profile intervals

While the transformation methods have the advantage of simplicity with the computations easily done by hand, a likelihood method uses the likelihood function directly to capture the skewness. Standard likelihood theory shows that the profile confidence interval is found by finding all values of \(N\) such that: \[-2 l(N) - 2 l(\widehat{N}) \le \chi^2_{\alpha}\]

However, in the conditional likelihood methods. \(N\) does NOT appear directly in the likelihood and so these methods are not relevant

3.3.3.8 Which method to choose?

All transformation methods should give similar results when the number of marks returned is reasonable. The methods could give quite different answers in cases when only a few marks are returned (say less than 20). However, in these cases, the differences in results among the confidence interval methods is small relative to the (unknown) biases and uncertainties from assumptions failures (such as non-mixing) that are still resident in the study.

The Petersen package created for this monograph used the logarithmic transformation to compute the confidence intervals for \(N\).

It should be noted that confidence intervals only capture the uncertainty in the estimate due to sampling – it does NOT capture uncertainty due to failure of assumptions, problems in the study, or other problems with the study.

3.4 Software

A simple internet search will find much software that can compute the Petersen estimator because the computations are so simple. However, these packages seldom take a coherent modeling strategy for the more complex models for Petersen studies. Similarly, the data structures seldom are consistent with modern software for capture-recapture studies (e.g., MARK).

Consequently, the Petersen R package has been developed to use a consistent methodology (conditional maximum likelihood estimation) with a consistent data frame work (capture histories). This package is available from the usual R repositories.

The VGAM (Yee et al., 2015) package also adopts the conditional likelihood approach for the closed population models with 2+ sample times, of which Petersen studies are simple cases. The use of this package is illustrated in Section 19.

The MARK program (and RMark which calls MARK) could be used for the simple Petersen studies, including the conditional likelihood approach. The use of this package is illustrated in Section 20.

3.5 Example - Estimating the number of fish in Rödli Tarn

Ricker (1975) gives an example of work by Knut Dahl on estimating the number of brown trout (Salmo truitta) in some small Norwegian tarns. Between 100 and 200 trout were caught by seining, marked by removing a fin (an example of a batch mark) and distributed in a systematic fashion around the tarn to encourage mixing. A total of \(n_1=109\) fish were captured, clipped and released, \(n_2=177\) fish were captured at the second occasion, and \(m_2=57\) marked fish were recovered.

The data are available in the data_rodli data frame in the Petersen package and can be accessed in the usual fashion:

library(Petersen)
data(data_rodli)

data_rodli
##   cap_hist freq
## 1       11   57
## 2       10   52
## 3       01  120

The data frame consists of (at least) two columns:

• a variable cap_hist with the two digit capture history (a character vector) -
• a variable freq that contains the number of fish in each capture history.

Additional columns can be present in the data frame for attributes of the fish such as sex, length etc.

The data frame can have a separate row for each fish (each with freq=1), or a summary as shown above.

You can construct the capture histories from the summary statistics as shown below:

cap_hist <- Petersen::n1_n2_m2_to_cap_hist( n1=109, n2=177, m2=57)
cap_hist

  cap_hist freq
1       10   52
2       01  120
3       11   57

3.5.1 Petersen estimate

We find the Petersen estimator of abundance using the conditional likelihood approach used in the Petersen package in two steps (the reason for the two steps will be explained later):

In the first step, the conditional-likelihood model is fit to the data:

rodli.fit.mt <- Petersen::LP_fit(data_rodli, p_model=~..time)

The LP_fit() function takes the data frame of capture histories seen earlier and a model for the capture probabilities (the p_model argument). The model for the capture probabilities can refer to any variable in the data frame (e.g., sex) or to several special variables such as ..time which refer to the two time periods. Some knowledge of how R sets up the design matrix given the data frame is helpful in deciding how to specify p_model in more complex situations involving stratification and is discussed later.

In the standard Petersen estimator, we allow the capture probabilities to vary across the two sampling events. Consequently, the p_model was specified as p_model=~..time.

A summary of the fit is given in a data frame

rodli.fit.mt$summary
##   p_model name_model   cond.ll n.parms nobs  method
## 1 ~..time p: ~..time -233.9047       2  229 cond ll

The value of the conditional likelihood, number of parameters in the conditional likelihood (just the two capture probabilities) and the number of observed fish (sum of the freq column) are also presented. Notice that in the conditional likelihood approach, the abundance is NOT a parameter in the likelihood and so is not counted in the model summary table.

In the second step, we obtain estimates of overall abundance using the LP_est() function and the N_hat argument. Here there is no stratification or other groupings of the data, so the formula for N_hat is ~1 indicating that we should find the estimated abundance for the entire population. (In later sections, we will obtain abundance estimates for individual strata as well).

rodli.est.mt <- Petersen::LP_est(rodli.fit.mt, N_hat=~1)

This again gives a data frame with the estimated abundance.

rodli.est.mt$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 338.4737 25.49646             0.95              logN
##   N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms nobs  method
## 1  292.0154  392.3232 ~..time p: ~..time -233.9047       2  229 CondLik

The estimated abundance is 338 (SE 25) fish computed as:

\(\widehat{N}=\frac{109 \times 177}{57}=338\).

The 95% confidence interval for \(N\) was computed using a logarithmic transformation is 292 to 392 fish.

The relative standard error (RSE) is found as \[RSE =\frac{SE}{estimate}=\frac{25.5}{338.5}=.075\]

is similar to the approximation that \[RSE \approx \frac{1}{\sqrt(marks~recovered)}=\frac{1}{\sqrt{57}}=.13\]

While the process of specifying a model for the capture probabilities and for the abundance estimate may seem a bit convoluted, its real power will become apparent when more complex cases involving stratification are discussed later.

3.5.2 Applying a bias correction

The Chapman modifications can be applied as outlined in Table 1. This can be implemented by adding a single “new” fish with capture history “11” to the existing data frame.

data(data_rodli)
rodli.chapman <- plyr::rbind.fill(data_rodli,
                    data.frame(cap_hist="11", 
                               freq=1, 
                               comment="Added for Chapman"))

rodli.chapman
##   cap_hist freq           comment
## 1       11   57              <NA>
## 2       10   52              <NA>
## 3       01  120              <NA>
## 4       11    1 Added for Chapman

Then this adjusted data is passed to the Petersen package and the two step procedure is again followed:

rodli.fit.mt.chap <- Petersen::LP_fit(rodli.chapman, p_model=~..time)
rodli.est.mt.chap <- Petersen::LP_est(rodli.fit.mt.chap, N=~1)

rodli.est.mt.chap$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 337.5861 25.02396             0.95              logN
##   N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms nobs  method
## 1  291.9364   390.374 ~..time p: ~..time -235.2889       2  230 CondLik

The bias-corrected estimate of the abundance is 338 (SE 25) fish.

3.5.3 A model with equal capture probabilities

It is also possible to fit a model where the capture probabilities are equal at both sampling events. This is specified by changing the specification for the model for \(p\) from \(p=\sim ..time\) to \(p=\sim 1\):

rodli.fit.m0 <- Petersen::LP_fit(data_rodli, p_model=~1)
rodli.est.m0 <- Petersen::LP_est(rodli.fit.m0, N_hat=~1)

This gives the estimates:

rodli.est.m0$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 358.7542 28.57733             0.95              logN
##   N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms nobs  method
## 1  306.8971  419.3738      ~1      p: ~1 -247.7207       1  229 CondLik

Now the estimated abundance is 359 (SE 29) fish.

However, a comparison of the model fits using AICc (Table 2), shows that this model has much less support than the traditional estimator with unequal capture probabilities:

rodli.aictab <- Petersen::LP_AICc( 
  rodli.fit.mt,
  rodli.fit.m0)

Table 2: Model comparison for the Rodli dataset

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~..time	-233.9047	2	229	471.86	0.00	1.00
p: ~1	-247.7207	1	229	497.46	25.60	0.00

Notice the \(N\) is not part of the conditional likelihood and so is not counted as a parameter.

In general, models with equal capture probability over sampling events are seldom sensible from a biological perspective.

4 Planning studies

Before determining the amount and distribution of effort to be allocated in a capture-recapture study, one must first decide what level of precision is required from the study. In many cases, a desired measure of the relative precision (relative standard error) is a goal of the study.

Some papers use the term coefficient of variation (\(cv\)) to refer to the relative standard error. The term relative standard error is preferred and the term \(cv\) is reserved for describing the variability of individual data values, i.e., \(cv = \frac{std~dev}{mean}\).

In turn, the relative standard error can be used to express the relative 95% confidence interval using the approximate rule that the relative confidence interval is about \(\pm 2\) relative standard errors.

Seber (1982, p. 64) gives three target levels of precision that are commonly used:

• For preliminary surveys where only a rough idea of the abundance is needed, the relative 95% confidence interval should be \(\pm 50\%\) of the estimated abundance with a corresponding relative standard error of 25%. For example, it would be sufficient for a preliminary survey to have results with an estimate of 50 ( \(SE\) 12) thousand fish with a 95% confidence interval of between 25 and 75 thousand fish.
• For accurate management work, the relative 95% confidence interval should be \(\pm 25\%\) of the estimated abundance with a corresponding relative standard error of 12.5%. For example, it would be sufficient for accurate management work to have results with an estimate of 50 ( \(SE\) 6) thousand fish, with a 95% confidence interval between 38 and 62 thousand fish.
• For careful scientific research, the recommended relative confidence interval is \(\pm 10\%\) of the estimated abundance with a corresponding relative standard error of 5%. For example, it would be sufficient for scientific work to have results with an estimate of 50 ( \(SE\) 2.5) thousand fish, with a 95% confidence interval of between 45 and 55 thousand fish.

If \(\widehat{N}\) is to be used for further computations, e.g. splitting the population by size categories or multiplied by average biomass to obtain an estimate of total biomass, then a high degree of precision is required so that the precision of the final answer is adequate.

As a rough rule of thumb, the relative precision (relative standard error) of the Petersen estimate is proportional to \(\frac{1}{\sqrt{m_2}}\), i.e., to the square root of the number of marks returned. This can be used for planning purposes. For example, if the initial abundance is about 50,000 fish, and if 5,000 fish are marked and 1,000 fish are examined for marks, then the approximate number of marks recaptured is: \[E[m_2] \approx 5,000 \times \frac{1,000}{50,000} = 100\] and the approximate relative standard error will be on the order of \[rse \approx \frac{1}{\sqrt{100}} = .1\] This will give a relative 95% confidence interval of about \(\pm 20\%\) which should be precise enough for management purposes. In general, the three levels of precision will require a certain number of marks returned as summarized in Table 3:

Table 3: Suggested minimum number of marks required to be returned to meet precision criteria

	Preliminary	Management	Scientific
95% relative ci	\(\pm\) 50%	\(\pm\) 25%	\(\pm\) 10%
Relative SE	25%	12%	5%
Required \(E[m_2]\)	16	64	400

The rule of thumb can be inverted to estimate the approximate fraction of the population that needs to be marked assuming equal sample sizes at both sampling occasions. In this case, \(E[m_2] \approx n^2/N \approx Nf^2\), where \(f\) is the sampling fraction. Consequently, if scientific precision is needed (a relative standard error of 5% with 400 marks needed to be recaptured), then a population of 10,000 will require \(10,000 f^2 = 400\) or \(f=.2=20\%\) of the population will needed to be sampled on both occasions.

While these rules of thumb should be sufficient for most purposes, one could actually use the estimated variances presented earlier. The workbook that accompanies this monograph has a worksheet where the scientist can enter various values of \(N\), \(n_1\), and \(n_2\) to see the approximate expected precision that will be obtained. For example, Figure 2, shows that for a abundance of about \(N=50,000\), sample sizes of \(n_1=3,000\) and \(n_2=1,000\) would give a relative standard error of about 12% and a relative 95% confidence interval of about \(\pm 25\%\) which should be sufficient for management purposes.

Figure 2: Illustration of sample size determination worksheet for population of \(N=5,000\) with \(n_1=1,000\) and \(n_2=3,000\) fish captured at the two sample occasions.

Robson and Regier (1964) gave nonograms which have been adopted and are presented in this monograph in Figure 3 to Figure 5. For example, using Figure 4 for a abundance of \(N=50,000\), the following pairs of sample sizes will be required to have the 95% relative confidence interval to be within \(\pm 25\%\): \[(1,000; 3,000), (800; 4,000), (300; 10,000)\] The charts are symmetric in \(n_1\) and \(n_2\), so that the same precision is obtained for \(N=50,0000\) with \((n_1=1,000; n_2=3,000)\) fish and \((n_1=3,000; n_2=1,000)\) fish.

Figure 3: Sample size nomogram for 10% relative 95% confidence intervals suitable for scientific work.

Figure 4: Sample size nomogram for 25% relative 95% confidence intervals suitable for management work.

Figure 5: Sample size nomogram for 50% relative 95% confidence intervals suitable for preliminary surveys.

While Robson and Regier (1964) and Seber (1982) have nomograms for abundances less than 200, these are rarely useful in practice.

The above spreadsheets and nonograms are sufficient for most planning purposes, but it should be kept in mind that these computations assume that the study will proceed perfectly and that all assumptions will be satisfied. This is rarely the case in most surveys, and so the required effort should be increased to account for potential problems down the road.

The above charts also assume that the cost of sampling per fish is equal at both sampling occasions. If the costs of sampling per fish differ at each sampling occasion, then Robson and Regier (1964) and Seber (1982, p. 69) show that the optimal allocation of effort between the two sampling occasions is the solution to:

\[\frac{c_1 n_1}{c_2 n_2} = \frac{N- n_2}{N- n_1} \tag{2}\]

where \(c_1\) and \(c_2\) are the costs of sampling per fish at the two occasions. In many cases, the sample sizes are negligible compared to the abundance so the right hand side of Equation 2 is 1, and the optimal allocation reduces to spending equal amounts of money at the two sampling occasions.

5 Assumptions and effects of violations

5.1 Introduction

The Petersen estimator makes a variety of assumptions and virtually no real study satisfies all of the assumptions. Consequently, a study of the effect of violations of assumptions upon the performance of the estimator is helpful in deciding how much credence should be placed upon the results.

In the sections that follow, the effect of violations of assumptions will be studied by substituting in the expected value of the statistics under the assumption violation. While this yields only an approximate answer to the effect of the violation, the results are sufficiently accurate to be useful in understanding.

In the workbook that accompanies this monograph, a specialized spreadsheet has been set up where the various effects (both individually and combined) can be studied without worrying about the hand computations (Section 5.7).

This is useful in order to see if violations of assumptions will lead to unacceptable bias. As suggested by Cochran (1977, p. 12-15), bias can typically be ignored if it is less than 10% of the standard error of the estimator.

There have been many papers that demonstrate how to modify the Petersen study to check for assumption violations and adjust the estimator – some of these are studied in future sections.

An ad-hoc adjustment can be made to the basic Petersen estimator if information is available about an assumption violation, e.g., an estimate of tag retention with accompanying SE. This is implemented in the LP_est_adjust() function as illustrated in Section 5.8.

5.2 Non-closure

The Petersen estimator assumes that the population of interest is closed. This means that no animals leave the population between the two sampling occasions through death or emigration, and no animals enter the population between the two sampling occasions through immigration or birth.

5.2.1 Immediate handling mortality

For many species, the act of catching and marking the fish traumatic and some fish suffer immediate mortality. In this case, the number of deaths is known. The study population should be reduced by these known mortalities and the resulting estimator should be conditional upon the actual number of fish released with tags rather than those captured.

The effect of marking upon subsequent mortality will be considered in Section 5.3.

5.2.2 Natural mortality or emigration

Suppose that natural mortality or emigration takes place between the two sampling occasions. These two types of non-closure are indistinguishable and shall be referred to as mortality in the remainder of this section.

What is the effect of this natural mortality if both the marked and unmarked fish have the same average survival probability between the two sampling occasions?

Let \(\phi\) represent the average survival probability between the two sampling occasions. Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[{n_2} \right] \approx \left[ {N p_1 \phi + N(1-p_1)\phi} \right] p_2\)
• \(E\left[{m_2} \right] \approx n_1 \phi p_2 = N p_1 \phi p_2\)

and

\[E[\widehat N] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times N\phi p_2 }}{{Np_1 \phi p_2 }} = N\]

i.e., the estimator remains essentially unbiased for the abundance at the time of the first sampling occasion.

In many cases, the mortality probability varies among groups (strata) of the population, e.g., different mortalities across different age groups. If the marked fish can be divided into groups (strata), a simple \(2 \times K\) contingency table can be constructed as shown in Table 4.

Table 4: Contingency table to test if mortalities are similar across different groups

	\(A\)	\(B\)	\(\ldots\)	\(K\)	Total
Number released	\(n_{1A}\)	\(n_{1B}\)	\(\ldots\)	\(n_{1K}\)	\(n_1\)
Recaptured	\(m_{2A}\)	\(m_{2B}\)	\(\ldots\)	\(m_{2K}\)	\(m_2\)
Not recaptured	\(n_{1A}-m_{2A}\)	\(n_{1B}-m_{2B}\)	\(\ldots\)	\(n_{1K}-m_{1K}\)	\(n_1-m_2\)

The usual \(\chi^2\) statistic is computed and is used to test if the product of survival and subsequent recapture (\(\phi_x p_{2x}\)) are equal across all groups (strata). If the hypothesis is rejected , then this MAY be evidence that the mortality probabilities are different in the subgroups – however it could also be evidence that the recapture probabilities are also unequal in the second sample across groups.

5.2.3 Immigration or births

Closure can also be violated by the addition of new animals, either by immigration from outside the study area or natural “births” (e.g., fish recruiting through growth). As both cases are indistinguishable, the term immigration will be used to refer to any increase in the population between the two sampling occasions.

Let \(\lambda\) be the rate of population increase between the two sampling occasions, i.e., an average of \(\lambda\) new fish enter the population per member of the original population. Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[{n_2} \right] \approx N \lambda p_2\)
• \(E\left[{m_2} \right] \approx n_1 \phi p_2 = N p_1 \phi p_2\)

and \[ \begin{array}{c} E[\widehat{N}] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times N\lambda p_2 }}{{Np_1 p_2 }} \\ = N\lambda = N_2 \\ \end{array} \] i.e., the Petersen estimator is approximately unbiased for the abundance at the second sampling occasion.

In many cases, recruitment varies across different groups of the population, e.g., smaller ages may tend to recruit as they get older between the two sample occasions.

If the fish recovered in the second sample can be divided into groups (strata), a simple \(2 \times K\) contingency table can be constructed as shown in Table 5.

Table 5: Contingency table to test for presence of recruitment among groups

	\(A\)	\(B\)	\(\ldots\)	\(K\)	Total
Number captured	\(n_{2A}\)	\(n_{2B}\)	\(\ldots\)	\(n_{2K}\)	\(n_2\)
Marked	\(m_{2A}\)	\(m_{2B}\)	\(\ldots\)	\(m_{2K}\)	\(m_2\)
Not marked	\(n_{2A}-m_{2A}\)	\(n_{2B}-m_{2B}\)	\(\ldots\)	\(n_{2K}-m_{1K}\)	\(n_2-m_2\)

The usual \(\chi^2\) statistic is computed and is used to test if the marked fraction are equals across all groups (strata). If the hypothesis is rejected , then this MAY be evidence of differential recruitment into the groups.

Seber (1982) has a non-parametric test for recruitment using length

5.2.4 Both immigration and mortality

Of course, both immigration and mortality can be occurring simultaneously.

As before, let \(\phi\) represent the average survival probability between the two sampling occasions, and let \(\lambda\) represent the net recruitment per individual alive at the start of the study before any mortality occurs. Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[ {n_2 } \right] \approx \left[ {N\left( {\lambda - 1} \right) + N\phi } \right]p_2\)
• \(E\left[{m_2} \right] \approx n_1 \phi p_2 = N p_1 \phi p_2\)

and \[ \begin{array}{c} E[\widehat{N}] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times \left[ {N\left( {\lambda - 1} \right) + N\phi } \right]p_2 }}{{Np_1 \phi p_2 }} = \frac{{N[\lambda - 1 + \phi ]}}{\phi } \\ \end{array} \]

i.e., the estimator now estimates the total number of animals ever alive over the course of the study being a combination of the initial abundance and the net number of recruits..

5.3 Marking has no effect

The physical act of marking a fish can be very traumatic to individual fish. Marking effects usually take two forms (both of which may be present in a study) (a) a change in subsequent survival or (b) a change in subsequent catchability.

5.3.1 Marking affects survival

Both an acute effect (immediate mortality after release) and a chronic effect (no immediate mortality but increased mortality between the two sampling occasions compared to unmarked fish) can be handled in the same way.

Let \(\phi\) represent the survival probability for unmarked fish between the two sampling occasions, and \(\phi '\) represent the survival probability for marked fish between the two sampling occasions.

Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[ {n_2 } \right] = \left[ {Np_1 \phi ' + N\left( {1 - p_1 } \right)\phi } \right]p_2\)
• \(E\left[ {m_2 } \right] = Np_1 \phi 'p_2\)

and \[ \begin{array}{c} E[\widehat{N}] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times \left[ {Np_1 \phi ' + N\left( {1 - p_1 } \right)\phi } \right]p_2 }}{{Np_1 \phi 'p_2 }} = N\left[ {\frac{\phi }{{\phi '}} + p_1 \left( {1 - \frac{\phi }{{\phi '}}} \right)} \right] \\ \end{array} \]

If the two survival probabilities are equal, then the ratios of survival probabilities are all equal to 1 and the estimator gives the number alive at the first sampling occasion as seen earlier. If the survival probability of marked fish is lower than the survival probability of unmarked fish, then the ratios are all greater than 1, and the Petersen estimator over estimates the number of fish. This is intuitive as increased mortality among the the marked fish leads to fewer marked fish being recaptured than expected and an inflation in the estimate.

In many studies \(p_1\) is small, and so the approximate relative bias is a function of the ratio of the two survival probabilities.

5.3.2 Marking affect catchability

The marking effect could affect subsequent catchability of the fish. This is often seen in small mammal studies where animals become trap happy or trap shy.

Let \(p_2\) represent the catchability at the second sampling occasion for unmarked fish, and \(p_2^{*}\) represent the catchability at the second sampling occasion for marked fish. Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[ {n_2 } \right] = Np_1 p_2 ' + N\left( {1 - p_1 } \right)p_2\)
• \(E\left[ {m_2 } \right] = Np_1 p_2 '\)

and

\[ \begin{array}{c} E[\widehat{N}] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times \left[ {Np_1 p_2 ' + N\left( {1 - p_1 } \right)p_2 } \right]}}{{Np_1 p_2 ' }} \\ = N\left[ {\frac{{p_2 }}{{p_2 ' }} + p_1 \left( {1 - \frac{{p_2 }}{{p_2 ' }}} \right)} \right] \\ \end{array} \]

If the two catchabilities are equal, the ratios are all one in the last expression, and the estimator is approximately unbiased. If fish become trap shy, then the ratios are both greater than unity, and the estimator again overestimates the abundance. Intuitively, trap shyness reduces the observed number of marks below what is expected and inflates the estimated abundance.

Again, if the capture probability at the first sampling event (\(p_1\)) is small, the relative bias can be approximated by the ratio of the subsequent catchabilities.

5.4 Tag loss

Tags can be lost for a variety of reasons, e.g.  breakage or tearing from the fish.

Let \(\rho\) represent the probability of retaining a tag between the two sampling occasions. Then

• \(E\left[{n_1} \right] \approx N p_1\)
• \(E\left[{n_2} \right] \approx N p_2\)
• \(E\left[ {m_2 } \right] = Np_1 \rho p_2\)

and

\[ E[\widehat{N}] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} = \frac{{Np_1 \times Np_2 }}{{Np_1 \rho p_2 }} = \frac{N}{\rho } \]

Intuitively, tag loss results in fewer tags being observed in the second sample than expected with a consequent positive bias in the estimated abundance. Using the rule of thumb from Cochran (1977), the bias resulting from tag loss can be “ignored” if

\[ \mathit{tag~loss~probability}= 1-\rho < \frac{0.1}{\sqrt{E[ m_T]}}\] where \(m_T\) is the number of marks actually recovered.

Tag loss can be detected by double tagging all or a fraction of the fish released (Section 10). This second tag can be a batch mark (e.g. a fin clip) or a second tag of the same or different tag material (e.g., a disc tag could be used for the second tag if the first tag was a spaghetti tag).

5.5 Tags not reported or overlooked

In some case, sampling in the second sample is done by anglers or volunteers. Tags can be overlooked. Generically, this problem is referred to a non-reporting of tags. The effects of non-reporting of tags is identical to that of tag-loss in Section 5.4 and so a detailed analysis will not be repeated here.

Tag non-reporting can be estimated and adjusted for in a number of ways

• A sub sample of the second sample can be reexamined for missed tag to estimate how many tags have been missed.
• Two classes of tags can be applied, one of which is expected to have a 100% reporting probability (e.g. high value reward tags).

5.5.1 Inspecting a sub-sample of “untagged” fish

Rajwani and Schwarz (1997) considered the situation where a sub-sample of fish that supposedly did not have tags is reinspected to count the number of missed tags. An “exact” analysis is found in the spreadsheet that ships with this package and is not detailed here.

Rajwani and Schwarz (1997) and the accompanying spreadsheet also derived at an optimal allocation of effort between the initial and secondary search to minimize the uncertainty in \(\widehat{N}\) for a given total budget.

An example of an empirical adjustment for non-reporting is found in Section 5.8.2.

5.5.2 Reward tags

In some cases, returns of tags are from anglers or other citizens and not all tags from recovered fish may be reported. To estimate the reporting probability, a second type of tag is also used (the “reward” tag) that offers a (monetary) incentive for its return. The incentive should be large enough to ensure that 100% of reward tags from recaptured fish are returned.

This type of study is exactly equivalent to a tag loss study and similar methods can be used. An example of the analysis of a reward tag study is found in Section 10.7.

5.6 Homogeneity in catchablity

Variable catchability is one of the major problem in capture-recapture methods. Heterogeneity in catchability has been divided into two categories – pure heterogeneity where catchability varies among fish but the relative catchability of individual fish among themselves does not change between sampling occasions, and variable heterogeneity where the catchability of individuals varying within sampling occasions and the relative catchability of individual may vary across sampling occasions.

5.6.1 Pure heterogeneity

In pure heterogeneity, fish vary in their catchability due to random chance or related to covariates such a sex or body length. However, the relative catchability of individuals compared to other individuals remains fixed over the course of the study. For example, males may be more catchable than females, and remain more catchable in both sampling occasions. Or nets may be used to select fish, and net selectivity is related to body length which doesn’t change much between sampling occasions.

To illustrate the effects of heterogeneity, suppose that there are two sub-populations with a fraction \(a\) in the first sub-population and \(1-a\) in the second sub-population. Let \(p_1\) and \(p_2\) represent the catchability of the first subpopulation at the two sampling occasions, and suppose that the second sub-population has catchabilities \(b p_1\) and \(b p_2\) respectively. The constant \(b\) represents the differential catchability among the two sub-populations at the two sampling occasions.

Then

• \(E\left[{n_1} \right] \approx N a p_1 + N (1-a) b p_1\)
• \(E\left[{n_2} \right] \approx N a p_2 + N (1-a) b p_2\)
• \(E\left[ {m_2 } \right] = N a p_1 p_2 + N (1-a) b p_1 b p_2\)

and \[ \begin{array}{c} E\left[ {\hat N} \right] \approx \frac{{E\left[ {n_1 } \right]E\left[ {n_2 } \right]}}{{E\left[ {m_2 } \right]}} \\ = \frac{{[Nap_1 + N(1 - a)bp_1 ][Nap_2 + N(1 - a)bp_2 ]}}{{Nap_1 p_2 + N(1 - a)bp_1 bp_2 }} \\ = N\frac{{\left[ {a + b(1 - a)} \right]^2 }}{{a + b^2 (1 - a)}} < N \\ {\textrm{ }} \\ \end{array} \]

In general, pure heterogeneity leads to a negative bias in the Petersen estimator. Intuitively, the more catchable fish are caught too often which leads to an increase in the observed number of marks compared to homogeneous catchable populations and a deflation of the estimator. Indeed, if certain fractions of the population are uncatchable (e.g., \(b=0\)) then the population estimates will always EXCLUDE the uncatchable segment and estimates only \(Na\). If the second sub-population has the same catchability as the first population \(a=1\), then there is no bias (as expected).

If the heterogeneity is related to a fixed categorical covariate (such as sex), then the bias can be removed by stratifying the population by categories and performing independent estimates for each category (see Section 6).

In many cases, heterogeneity is related to a continuous covariate such a body length and induced by net selectivity. This continuous covariate can be used to model the catchability. Alternatively, Chen and Lloyd (2000) developed a non-parametric method to account for this type of heterogeneity – see Section 7.3 for details.

5.6.2 Variable heterogeneity

5.6.2.1 General heterogeneity

Both the cases above (pure heterogeneity and changing heterogeneity) can be subsumed into a general expression for the bias introduced by heterogeneity.

This can be generalized to a continuous distribution of catchabilities, say as a function of body length. Junge (1963) and Seber (1982, p. 86) show that the relative bias (\(RB=\frac{E[\widehat{N}]-N}{N}\)) can be approximated by: \[RB \approx - C \left( {p_{1j},p_{2j}} \right) \times \frac{\sqrt {V(p_{1j}) V(p_{2j}) }}{E\left[ {p_{1j} p_{2j}} \right]}\] where \(C(\cdot,\cdot)\) is the correlation of catchabilities between sampling occasions, \(V(\cdot)\) are the variability of the catchabilities at each sampling occasion, all taken over the individuals (\(j\)).

If all animals are equally catchable at either sampling occasion (e.g., a random sample at either sampling occasion), then the \(p_j\) are constant, the correlation is zero because any random variable has a 0 correlation with a constant. There is no bias.

If the catchabilities vary among individuals but the catchability in the second sample does not depend upon the catchability in the first sample, the correlation is again zero and there is no bias. This is reason for recommending that different sampling methods be used a each sampling occasion.

If the heterogeneity is related to a fish covariate such as size and the same sampling methods are used in both sampling occasions, then a positive correlation exists between the catchabilities and a negative bias in the estimate occurs.

Trap shyness leads to a negative correlation between the two capture probabilities, and a positive bias as seen earlier.

Junge(1963) and Seber(1982, p. 86) also examined how extreme the bias could be in the special case where \(p_{2j}=bp_{1j}\), i.e., pure heterogeneity among animals. In this case the correlation is 1, and the relative bias simplifies to: \[RB \approx - \frac{V(p_{1j})}{E\left[ {p_{1j}^2} \right]}\] This can be used to approximate the extent of the bias introduced by pure heterogeneity.

Expand on this here more, e.g. assume a beta distribution for p1 with various SD.

5.7 Spreadsheet to investigate impact of assumption violations

It is rare that a simple violation of assumptions will be occurring. The effect of multiple violations can be investigated using the spreadsheet supplied with this monograph.

The spreadsheet is set up to accommodate up to 4 segments of the population (e.g., strata), and has columns for the various assumption violations. The baseline condition is set up using guessestimates for the abundance and catchability probability and, as expected, no bias is seen in the estimates (Figure 6):

(a) Parameter values

(b) Estimated bias

Figure 6: Baseline conditions for assessing impact of assumption violations

Now it is straight forward to investigate differences in catchability catchability and mortality between marked and unmarked fish. Suppose that marked fish are slightly less catchable at the second sampling event (\(p_2^{marked}=.016\); \(p_2^{unmarked}=.017\)) and that marked fish have a survival probability of 0.95. The resulting bias is around 17% (Figure 7):

(a) Parameter values

(b) Estimated bias

Figure 7: Impact of violations of assumptions

If heterogeneity is related to a fixed attribute, such as sex, the parameter values now are entered into 2 (or more) rows of the table. For examples, suppose we have a population with a 50:50 sex ratio; females are less catchable at the first event, and more catchable at the second event. The approximate bias is around 20% (Figure 8):

(a) Parameter values

(b) Estimated bias

Figure 8: Impact of heterogeneity associated with sex.

Finally, if heterogeneity is associated with geographic or temporal movement, a simple example allowing for a combination of 2 tagging strata (N vs S) and 2 recapture strata (N vs S) can be examined. Here we divide the population in to 4 categories corresponding to the 4 possible pairs of movements, and specify suitable values for the catchabilities. The estimated bias is around 5% (Figure 9):

Parameter values

Estimated bias

Figure 9: Impact of heterogeneity associated with geographic stratification.

Generally speaking, two tagging x two recapture strata will be sufficient to determine the approximate size of any bias.

5.8 Empirical adjustments to existing estimates

While it is possible to obtain a corrected point estimate using the analytical analysis above, finding the SE of the adjust estimate is more complex, especially if several adjustment factors are involved.

Consequently, and empirical adjustment can be obtained using the LP_est_adjust() function that takes the estimates (and SE) of abundance and estimates of the adjustment factor (and SE) and simulates the impact of the adjustments. A log-normal distribution is assumed for the estimates of abundance, and a distribution for each individual adjustment factor is determined using the corresponding estimate and SE. Once the effects of the combined adjustments is simulated, the distribution of the abundance estimates is back-transformed and the adjusted SE (and confidence intervals) is determined. This process is analogous to what happens in a Bayesian analysis.

5.8.1 Adjusting for tag loss

5.8.1.1 Adjusting Rodli estimates for tag loss

Suppose we wish to adjust the Rodli estimates for tag loss. Recall the estimated abundance was 338 (SE 25) fish.

Suppose that the empirical estimate of tag retention is 0.90 (SE .05). The adjusted estimate is found as:

set.seed(23432)
rodli.est.mt.adjust <- LP_est_adjust(
              rodli.est.mt$summary$N_hat, rodli.est.mt$summary$N_hat_SE,
              tag.retention.est=0.90, tag.retention.se=0.05,
              )

A comparison of the original and adjusted estimates of the abundance are:

rodli.est.mt.adjust$summary

  N_hat_un N_hat_un_SE N_hat_adj N_hat_adj_SE N_hat_adj_LCL N_hat_adj_UCL
1 338.4737    25.49646  305.7388     28.62206      250.8642      364.0012

5.8.1.2 Example from a double tagging study

In Section 10.4.2, a double tagging study was conducted, and estimates of abundance and and the tag retention probability were obtained. We will re-analyze this data, using only 1 tag and then do the empirical adjustment (assuming we know the tag retention probability),

First we get the double tag data and remove information from the second tag (it doesn’t matter that some revised histories are duplicated).

data(data_sim_tagloss_twoD)

data_one_tag <- data_sim_tagloss_twoD
data_one_tag$old_cap_hist <- data_one_tag$cap_hist
data_one_tag$cap_hist <- paste0(substr(data_one_tag$cap_hist,1,1),
                                substr(data_one_tag$cap_hist,3,3))

This gives the “single tag” data:

  cap_hist freq old_cap_hist
1       01  879         0010
2       10  225         1000
3       11   14         1010
4       10  666         1100
5       10   21         1101
6       11    7         1110
7       11   37         1111

We now fit the regular Petersen estimator and get the estimates that are now biased because of tag loss:

data_one_tag.fit <- Petersen::LP_fit(data_one_tag, p_model=~1)
data_one_tag.est <- Petersen::LP_est(data_one_tag.fit)
data_one_tag.est$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 15675.21 1933.056             0.95              logN
  N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms nobs  method
1   12309.6  19961.03      ~1      p: ~1 -1499.301       1 1849 CondLik

And we make the empirical adjustment based on the results from the tag loss study:

data_one_tag.est.adj <- Petersen::LP_est_adjust(
                             data_one_tag.est$summary$N_hat, 
                             data_one_tag.est$summary$N_hat_SE,
                             
                             tag.retention.est=.64, 
                             tag.retention.se =.06)

giving:

data_one_tag.est.adj$summary

  N_hat_un N_hat_un_SE N_hat_adj N_hat_adj_SE N_hat_adj_LCL N_hat_adj_UCL
1 15675.21    1933.056  10099.55     1562.258      7339.145      13413.14

The estimate is similar to that in Section 10.4.2, but of course the SE is larger because the information from the other tags is not longer available.

5.8.2 Overlooked tags/Non-reporting of tags

Rajwani and Schwarz (1997) considered the situation where a sub-sample of fish that supposedly did not have tags is reinspected to count the number of missed tags. The data on males is taken from Rajwani and Schwarz (1997)

As fish return to their spawning sites, \(n_1=1510\) are captured using seine nets. A Petersen disk tag is attached and the fish is released. After spawning, the fish die and the carcasses are often washed onto the banks of the spawning area. Survey teams walk along the banks looking for carcasses. When a carcass is found, it is examined for a tag. After enumeration, all tags are cut from the carcasses, and those carcasses are removed from the study area by cutting them into two with a machete and returning them to the river. Untagged carcasses are left where found.

A total of \(n_2=45595\) carcasses are examined and \(m_2=279\) marks are observed. Later in the season, a second team examines some of those carcasses identified as being without tags to check for tags missed in the initial survey. A total of \(n_3=8462\) carcasses are reexamined (subsampled) and \(m_3=6\) new tags are found.

The data and capture histories are:

  cap_hist  freq
1       10  1231
2       11   279
3       01 45316

We start with the standard conditional-likelihood Petersen estimate:

data_overlook.fit <- Petersen::LP_fit(data_overlook, p_model=~..time)
data_overlook.est <- Petersen::LP_est(data_overlook.fit)
data_overlook.est$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 246768.4 13298.26             0.95              logN
  N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms  nobs  method
1  222033.5  274258.7 ~..time p: ~..time -7393.831       2 46826 CondLik

This estimate is biased upwards because of the non-reporting of tags. Rajwani and Schwarz (1997) give the analytical expressions for the estimate of the reporting probability: \[\widehat{\rho} = \frac{m_2}{m_2+ n_{01}\frac{m_3}{n_3}}=\frac{279}{279+ 45316\frac{6}{8462}}\]

We estimate the uncertainty in the reporting probability empirically:

# estimate the reporting rate
rho = m2/(m2+ (n2-m2)*m3/n3)

rho.sim <- m2/(m2+ (n2-m2)*(rbeta(10000,m3,n3-m3)))
rho.se = sd(rho.sim)

This gives an estimated reporting probability of 0.897 (SE 0.037).

These values are used to adjust the previous estimate

data_overlook.est.adj <- Petersen::LP_est_adjust(
                             data_overlook.est$summary$N_hat, 
                             data_overlook.est$summary$N_hat_SE
                             ,
                             tag.reporting.est=rho, 
                             tag.reporting.se =rho.se)
data_overlook.est.adj$summary

  N_hat_un N_hat_un_SE N_hat_adj N_hat_adj_SE N_hat_adj_LCL N_hat_adj_UCL
1 246768.4    13298.26  221357.3     14977.76      192665.1      250574.5

These results are very similar to those in Rajwani and Schwarz (1997) of \(\widehat{N}\)=221,284 (SE 15,068) fish.

5.8.3 Adjusting tags available using Bayesian methods

• refer to the Nass river study

6 Accounting for heterogeneity I - fixed discrete strata

Heterogeneity is likely the most common reason for extensive bias in the Petersen estimator. Pure heterogeneity, i.e., some fish are always more catchable than other fish, leads to a negative bias in the estimated abundance. Heterogeneity that varies among fish and between the two sample times can lead to positive or negative bias depending upon the correlation of the heterogeneity over the animals between the two sample times.

A common way to correct for biases caused by heterogeneity is stratification where animals are separated in to groups by a measured covariate. There are four types of covariates that are commonly measured in (roughly) increasing order of complexity of analysis:

• fixed, individual categorical covariates such as sex.
• fixed, individual continuous covariates such as length in short studies.
• changing, individual, categorical covariates such as location or timing of capture at each sampling event.
• changing, individual continuous covariates such as length in long studies.

This section demonstrates the analysis of fixed, individual covariates such as sex.

6.1 Fixed individual categorical covariates such as sex

Very often a simple fixed categorical covariate is measured on individual fish such as sex at both sampling occasions. This device can also be used with continuous covariates (such as length) that do not change much over the course of a season if the continuous covariate is broken into distinct classes (e.g., length classes) as shown in Section 6.3.

A key assumption for this section is that fish do not change categories between the two sampling intervals.

6.1.1 Test for equal marked fractions

As a first step, a simple contingency table test can be used to examine if there is statistical evidence of differential catchability either in the first or second samples. Recall that the estimated catchabilities are found as \(\widehat{p}_1 = \frac{m_2}{n_2}\) and \(\widehat{p}_2 = \frac{m_2}{n_1}\). So, if there are \(K\) classes, a contingency table to test the equality of catchability at the first sample is equivalent to the test of presence of recruitment among groups found in Table 5 and is a test of equal marked fractions across the strata.

The usual \(\chi^2\) statistic is computed and is used to test if the marked fraction (the catchability at time \(1\)), are equal across all strata. If the hypothesis is rejected , then this MAY be evidence of differential catchability at time \(1\).

6.1.2 Test for equal catchability

Similarly, the contingency table found in Table 4 can be used to test if the recapture probabilities at the second sampling occasion are equal.

6.1.3 Stratifed models

If there is evidence that catchabilities may differ among the strata at least one sampling occasion, the simplest strategy is to compute separate Petersen estimates for each stratum and then combine the estimates. For example, if the population is stratified by sex and no assumption is made about the sex ratio at each sampling occasion, nor about equality of catchability at the sampling occasions, application of the previous methods lead to two estimates of abundance, one for each sex, and their associated standard error.

The combined estimate of abundance is found as: \[\widehat{N}_{combined}= \widehat{N}_f + \widehat{N}_m\] and the \(SE\) of the combined estimator is found as: \[se(\widehat{N}_{combined})= \sqrt{se(\widehat{N}_f)^2 + se(\widehat{N}_m)^2}\] The extension to more than two categories is straightforward.

However, this strategy could be inefficient, if the catchability differs among the strata only at a single sampling occasion. For example, the catchability of both sexes may be the same at sample time \(1\), but differ at sample time \(2\).

There are four possible general models that could be fit. There are also several possible models where only some of the strata have equal catchability at either/both of the sampling occasions. The general theory presented here will also accommodate these cases.

• Complete homogeneity: \(p_{1A}=p_{1B}=\ldots=p_{1K}\) and \(p_{2A}=p_{2B}=\ldots=p_{2K}\).
• Homogeneity at time 1 only: Only \(p_{1A}=p_{1B}=\ldots=p_{1K}\).
• Homogeneity at time 2 only: Only \(p_{2A}=p_{2B}=\ldots=p_{2K}\).
• Complete heterogeneity: No equality at either sampling occasion.

The Akiake Information Criterion (AIC; Akaike, 1973) paradigm is a preferred method of dealing with multiple models for the same data. Burnham and Anderson (2002) provides a detailed reference on the use of this methodology. In brief, this paradigm asserts that none of the models fit to data are the “truth” (do you really believe that all individual of the same sex have exactly the same catchability). Rather all models are approximation to reality and several models may approximate reality almost equally well. The AIC statistic is computed for each model and is a measure of fit and a penalty for the number of parameters used to fit the model. The AIC statistic can be used to derive model weights which are a measure of relative strength in explaining the data among the competing models. A weighted average of the estimates from the competing model can be computed and the \(SE\) of this estimate can incorporate both the individual precision from each of the model plus a measure of model uncertainty. For example, if the various models give vastly different estimates of the population abundance, then the model uncertainty about the population abundance will be large. Conversely, if the various models agree to a great extent in their estimate, then the uncertainty in the estimate due to model choice is small.

In some cases, additional information can, and should be used to improve precision. For example, in many species, the sex ratio in the population is known from other surveys or is assumed to be 50:50. I am unaware of any previous work on incorporating knowledge of the the stratum population ratios into the estimation framework.

6.2 Example: Northern Pike - simple stratification

In 2005, the Minnesota Department of Natural Resources conducted a tagging study to estimate the number of northern pike (Esox lucius) in Mille Lacs, Minnesota. Briefly, approximately 7,000 fish were sexed and tagged on their spawning grounds in the spring and a summer gillnet assessment captured about 1,000 fish. Complete details are in Bruesewitz and Reeves (2005).

Fish were double tagged, but an analysis using a tag-retention model showed that tag loss was negligible (Section 10.6).

Each fish has its own individual history. The sex and length (inches) of the fish at the time of capture are also recorded. The first few records are:

data(data_NorthernPike)
head(data_NorthernPike)
##   cap_hist length Sex freq
## 1       01  23.20   M    1
## 2       01  28.89   F    1
## 3       01  25.20   M    1
## 4       01  22.20   M    1
## 5       01  25.00   M    1
## 6       01  24.70   M    1

6.2.1 Summary statistics by sex

The summary statistics by sex are found in Table 6.

Table 6: Summary statistics for Northern Pike capture-recapture study

Sex	n1	n2	m2	P(recapture)	Marked fraction
F	4,045	613	89	0.022	0.145
M	2,777	527	68	0.024	0.129
ALL	6,822	1,140	157	0.023	0.138

The recapture probabilities are similar across the two sexes as are the marked fractions. However, with a large sample size, small differences in capture probabilities may be detected.

The analysis will be done using the capture histores for individual fish, but you can construct the capture histories from the summary statistics as shown below:

cap_hist <- n1_n2_m2_to_cap_hist(n1=c(4045,2777), 
                                 n2=c(613,527), 
                                 m2=c(89,68),
                                 strata=c("F","M"),stratum_var="Sex")
cap_hist

  cap_hist freq Sex
1       10 3956   F
2       10 2709   M
3       01  524   F
4       01  459   M
5       11   89   F
6       11   68   M

6.2.2 Test for equal marked fractions by sex

The contingency table to test for equal catchability at sampling occasion 1 (indicated by the marked fraction seen at the second occasion) is computed using the LP_test_equal_mf() function to test for equal marked fractions:

nop.equal.mf.sex <- LP_test_equal_mf(data_NorthernPike, "Sex") 

This gives the summary table in Table 7.

Table 7: Contingency table for testing equal marked fraction

	Number of fish		Proportions
status	F	M	F	M
Not seen at t1	524	459	0.855	0.871
Recaptured	89	68	0.145	0.129

and the resulting chi-square test output is:

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  tab
## X-squared = 0.4942, df = 1, p-value = 0.4821

The \(\chi^2\) test \(p\)-value for equal marked fraction is p = 0.482 and so there is no evidence of differential marked fractions.

6.2.3 Test for equal recapture probabilities by sex

The contingency table to test for equal recapture probability from fish tagged at sampling occasion 1 is computed using the LP_test_equal_recap() function:

nop.equal.recap.sex <- LP_test_equal_recap(data_NorthernPike, "Sex") 

This gives the summary table in Table 8.

Table 8: Contingency table for testing equal recapture probability

	Number of fish		Proportions
status	F	M	F	M
Never seen	3,956	2,709	0.978	0.976
Recaptured	89	68	0.022	0.024

and the results of the \(\chi^2\) test are:

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  tab
## X-squared = 0.34826, df = 1, p-value = 0.5551

The \(p\)-value for equal recapture probability is p = 0.555 and so there is no evidence of differential recapture probabilities between the two sexes.

6.2.4 Fitting multiple models by sex

While these two tests indicate no evidence of differential catchability at either sampling occasion, they do not necessarily indicate that the two sexes can be completely pooled when computing the Petersen estimates.

Several models can be fit to the NorthernPike data:

• Completely pooled Petersen. Capture-probabilities at each sampling event do not depend on Sex. This is equivalent to fitting a simple Petersen model ignoring sex.
• Completely stratified-Petersen. Capture-probabilities at each sampling event depend on Sex. This is equivalent to fitting two separate Petersen models, one for each sex.
• Pure heterogeneous models where the probability of capture varies between the two sexes, but is consistently the same across sampling events. This is equivalent to an additive model between sex and time.

These models were fit to the northern pike data:

data(data_NorthernPike)

# Fit the various models

nop.fit.time       <- Petersen::LP_fit(data_NorthernPike, p_model=~..time)
nop.fit.sex.time   <- Petersen::LP_fit(data_NorthernPike, p_model=~-1+Sex:..time)
nop.fit.sex.p.time <- Petersen::LP_fit(data_NorthernPike, p_model=~Sex+..time) 

# Fit models where the p(capture) is equal at t1 or t2 but not both.
# This is intermediate between the ~..time and ~..time:Sex models

nop.fit.eq.t1       <- Petersen::LP_fit(data_NorthernPike, 
                              p_model=~-1+I(as.numeric(..time==1))+
                                          I(as.numeric(..time==2)):Sex)
                                    
nop.fit.eq.t2       <- Petersen::LP_fit(data_NorthernPike, 
                              p_model=~-1+I(as.numeric(..time==2))+
                                          I(as.numeric(..time==1)):Sex)

Notice the formula for the p_model to force the capture-probabilities to be equal at the first or second sampling event and to vary by sex for the other sampling event. The I() notation in the model formula indicates that the interior expression is to evaluated first before being using in the model. In this case, we create an indicator variable if the sampling event is the first or second event.

We can now rank these models in terms of AICc (Table 9):

# compare the various models

nop.sex.aictab <- LP_AICc(
        nop.fit.time,
        nop.fit.sex.time,
        nop.fit.sex.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2)  

Table 9: Comparison of models fit to the Northern Pike data

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):Sex	-3,696.051	3	7,805	7,398.11	0.00	0.42
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):Sex	-3,696.139	3	7,805	7,398.28	0.18	0.38
p: ~-1 + Sex:..time	-3,695.826	4	7,805	7,399.66	1.55	0.19
p: ~..time	-3,702.341	2	7,805	7,408.68	10.58	0.00
p: ~Sex + ..time	-3,702.228	3	7,805	7,410.46	12.35	0.00

The AIC weight indicate high support for either model where the capture probabilities are equal either at the first or the second but not both sampling occasions. The model that involves complete pooling over both sexes (pooled Petersen) is given a very low weight (less than 1%). The model that has completely separate estimates for males and females is given a lower weight (only about 19%) as well.

We can now extract the estimates of the overall abundance from the models and obtain the model averaged values as shown in Table 10:

# extract the estimates of the overall abundance

nop.sex.ma.N_hat_all<- LP_modavg(
        nop.fit.time,
        nop.fit.sex.time,
        nop.fit.sex.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, N_hat=~1)  

Table 10: Model averaged estimates of overall abundance for Northern Pike

N_hat_f	N_hat_rn	Modnames	AICcWt	Estimate	SE
~1	(Intercept)	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):Sex	0.42	49,536	3,629
~1	(Intercept)	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):Sex	0.38	49,536	3,629
~1	(Intercept)	p: ~-1 + Sex:..time	0.19	49,382	3,616
~1	(Intercept)	p: ~..time	0.00	49,536	3,629
~1	(Intercept)	p: ~Sex + ..time	0.00	49,596	3,643
		Model averaged		49,506	3,627

Notice that the estimates (and SE) for three of the models are identical – the pooled Petersen estimator (4th model), and the two model where the capture-probabilities are the same at either sampling event. The latter two models are one of the cases where the pooled-Petersen is unbiased, i.e., the capture-probabilities are homogeneous at either of the sampling events.

If only the model where each sex is modeled independently was used, the estimated abundances would have been 49,382 ( \(SE\) 3,616) fish. In the models where the effect of sex was ignored either completely or at either of the sampling occasions, the estimated abundance would have been 49,536 ( \(SE\) 3,629) fish. In this case, this extra work in trying to model the catchabilities did not lead to estimates that were very different than this most general model.

So what is the advantage of the first two models vs. the pooled-Petersen. It is now possible to obtain estimates of the abundance of the two sub-populations. In the traditional pooled-Petersen estimator, it is not easy to get estimates of the sub-population, but this can also be done using the conditional likelihood approach followed by the Horvitz-Thompson estimator (Table 11).

# extract the estimates of the abundance for each sex
nop.sex.ma.N_hat_sex<- LP_modavg(
        nop.fit.time,
        nop.fit.sex.time,
        nop.fit.sex.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, N_hat=~-1+Sex)  

Table 11: Model averaged estimates of abundance for each sex for Northern Pike

N_hat_f	N_hat_rn	Modnames	AICcWt	Estimate	SE
~-1 + Sex	SexF	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):Sex	0.42	26,814	2,048
~-1 + Sex	SexF	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):Sex	0.38	29,338	2,171
~-1 + Sex	SexF	p: ~-1 + Sex:..time	0.19	27,860	2,700
~-1 + Sex	SexF	p: ~..time	0.00	28,998	2,139
~-1 + Sex	SexF	p: ~Sex + ..time	0.00	29,828	2,859
		Model averaged		27,993	2,506
~-1 + Sex	SexM	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):Sex	0.42	22,722	1,823
~-1 + Sex	SexM	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):Sex	0.38	20,197	1,499
~-1 + Sex	SexM	p: ~-1 + Sex:..time	0.19	21,522	2,406
~-1 + Sex	SexM	p: ~..time	0.00	20,538	1,526
~-1 + Sex	SexM	p: ~Sex + ..time	0.00	19,768	2,135
		Model averaged		21,513	2,160

Notice how estimates of abundance are requested for each sex. Now the estimates of the sub-population abundances differ among the models even though the estimates overall population abundances are equal

6.3 Example: Northern pike stratified by length class

We return to the Northern Pike example. The length of the fish was also measured and the sampling events are close enough that the change in length between the two sampling events is negligible.

6.3.1 Distribution of length in captured fish

A histogram of the distribution of length in the handled fish is show in Figure 10.

Figure 10: Histogram of length measurements for Northern Pike

It appears that female fish tend to be larger than male fish.

6.3.2 Summary statistics by length class

We start by classifying the length into length classes of 0:20 in, 20:25 in, 25:30 in, 30:35 in; and 35+ in. Summary statistics are presented in @bl-nop-lengthclass-sumstat

Table 12: Summary statistics by length class

length.class	n1	n2	m2	P(recapture)	Marked fraction
00-20	569	22	1	0.002	0.045
20-25	2,238	371	41	0.018	0.111
25-30	2,120	537	86	0.041	0.160
30-35	1,254	173	25	0.020	0.145
35+	641	37	4	0.006	0.108
ALL	6,822	1,140	157	0.023	0.138

There appears to differences in recaptured probability by length class peaking around the 25-30 inch class, but the marked-fractions appear to be similar. Notice the small number of recaptures in the first and last length classes – these strata likely should be pooled with the other strata if a fully-stratified model is used to avoid the small sample biases.

6.3.3 Test for equal marked fractions by length class

The contingency table to test for equal catchability at sampling occasion 1 (indicated by the marked fraction seen at the second occasion) is constructed as illustrated in Table 13.

nop.equal.mf.length.class <- LP_test_equal_mf(data_NorthernPike,
                                              "length.class") 

Table 13: Contingency table for testing equal marked fraction

	Number of fish					Proportions
status	00-20	20-25	25-30	30-35	35+	00-20	20-25	25-30	30-35	35+
Not seen at t1	21	330	451	148	33	0.955	0.889	0.840	0.855	0.892
Recaptured	1	41	86	25	4	0.045	0.111	0.160	0.145	0.108

and the resulting chi-square test output is:

## 
##  Pearson's Chi-squared test
## 
## data:  tab
## X-squared = 6.505, df = 4, p-value = 0.1645

The \(p\)-value for equal marked fraction is p = 0.164 and so there is no evidence of differential marked fractions among the length classes.

6.3.4 Test for equal recapture probabilities by length class

The contingency table to test for equal recapture probability from fish tagged at sampling occasion 1 is constructed as illustrated in Table 14.

nop.equal.recap.length.class <- LP_test_equal_recap(data_NorthernPike, 
                                                    "length.class") 

Table 14: Contingency table for testing equal recapture probability

	Number of fish					Proportions
status	00-20	20-25	25-30	30-35	35+	00-20	20-25	25-30	30-35	35+
Not recaptured	568	2,197	2,034	1,229	637	0.998	0.982	0.959	0.980	0.994
Recaptured	1	41	86	25	4	0.002	0.018	0.041	0.020	0.006

and the resulting chi-square test output is:

## 
##  Pearson's Chi-squared test
## 
## data:  tab
## X-squared = 51.225, df = 4, p-value = 2.003e-10

The \(p\)-value for equal recapture probability is p < .001 and so there appears to be good evidence of differential recapture probabilities among the length classes. However, some of the counts in some cells are quite small and so the p-value may be too small and a Fisher Exact test may be preferable.

## 
##  Fisher's Exact Test for Count Data with simulated p-value (based on
##  2000 replicates)
## 
## data:  tab
## p-value = 0.0004998
## alternative hypothesis: two.sided

The \(p\)-value is also small, so there is evidence of a difference in the recapture probabilities by length class.

6.3.5 Fitting multiple models by length class

We fit the suite of models similar to those when stratifying by sex as shown in Table 15.

# Fit the various models

nop.fit.time       <- Petersen::LP_fit(data_NorthernPike, 
                                       p_model=~-1+..time)
nop.fit.length.class.time   <- Petersen::LP_fit(data_NorthernPike,
                                       p_model=~-1+length.class:..time)
nop.fit.length.class.p.time <- Petersen::LP_fit(data_NorthernPike,
                                       p_model=~length.class+..time) 

# Fit models where the p(capture) is equal at t1 or t2 but not both.
# This is intermediate between the ~..time and ~..time:Sex models

nop.fit.eq.t1       <- Petersen::LP_fit(data_NorthernPike, 
                                      p_model=~-1+I(as.numeric(..time==1))+
                                                  I(as.numeric(..time==2)):length.class)
                                    
nop.fit.eq.t2       <- Petersen::LP_fit(data_NorthernPike, 
                                        p_model=~-1+I(as.numeric(..time==2))+
                                                    I(as.numeric(..time==1)):length.class)

We can now rank these models in terms of AICc (Table 15):

# compare the various models

nop.sex.aictab <- LP_AICc(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2
        )  

Table 15: Comparison of models fit to the Northern Pike data stratifying by length class

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	-3,596.138	6	7,805	7,204.29	0.00	0.62
p: ~-1 + length.class:..time	-3,592.615	10	7,805	7,205.26	0.97	0.38
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	-3,621.220	6	7,805	7,254.45	50.16	0.00
p: ~length.class + ..time	-3,678.014	6	7,805	7,368.04	163.75	0.00
p: ~-1 + ..time	-3,702.341	2	7,805	7,408.68	204.40	0.00

The AIC weight indicate high support for either model where the capture probabilities are equal at the first sampling event (equal marked fraction) and secondarily, to a model with full stratification at both sampling events.

We can now extract the estimates of the overall abundance from the models and obtain the model averaged values as shown in Table 16:

# extract the estimates of the overall abundance

nop.length.class.ma.N_hat_all<- LP_modavg(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, N_hat=~1)  

Table 16: Model averaged estimates of overall abundance for Northern Pike when stratifying by length class

N_hat_f	N_hat_rn	Modnames	AICcWt	Estimate	SE
~1	(Intercept)	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	49,536	3,629
~1	(Intercept)	p: ~-1 + length.class:..time	0.38	60,614	13,039
~1	(Intercept)	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	49,536	3,629
~1	(Intercept)	p: ~length.class + ..time	0.00	93,218	39,716
~1	(Intercept)	p: ~-1 + ..time	0.00	49,536	3,629
		Model averaged		53,754	10,091

Notice that the estimates (and SE) for three of the models are identical – the pooled Petersen estimator, and the two model where the capture-probabilities are the same at either sampling event. The latter two models are one of the cases where the pooled-Petersen is unbiased, i.e., the capture-probabilities are homogeneous at either of the sampling events. However, there really is only support for the model with equal capture probability at the first sampling event.

Notice that the estimate for the fully stratified Petersen is quite large but with a very large standard error. This is likely an artefact of the small number of recaptures. It is left as an exercise for the reader to refit the models, pooling the first and last length classes with the second or second-to-last classes to avoid the small sample bias problem.

Finally, here are the estimates of abundance by length class using the the Horvitz-Thompson estimator (Table 17).

# extract the estimates of the abundance for each sex
nop.length.class.ma.N_hat_length.class<- LP_modavg(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, N_hat=~-1+length.class)  

Table 17: Model averaged estimates of abundance for each length class for Northern Pike

N_hat_rn	Modnames	AICcWt	Estimate	SE
length.class00-20	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	4,146	345
length.class00-20	p: ~-1 + length.class:..time	0.38	12,518	12,220
length.class00-20	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	1,481	210
length.class00-20	p: ~length.class + ..time	0.00	39,449	39,004
length.class00-20	p: ~-1 + ..time	0.00	3,745	306
	Model averaged		7,334	8,571
length.class20-25	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	16,324	1,224
length.class20-25	p: ~-1 + length.class:..time	0.38	20,251	2,955
length.class20-25	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	16,577	1,374
length.class20-25	p: ~length.class + ..time	0.00	20,014	2,849
length.class20-25	p: ~-1 + ..time	0.00	16,298	1,218
	Model averaged		17,819	2,809
length.class25-30	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	15,306	1,136
length.class25-30	p: ~-1 + length.class:..time	0.38	13,238	1,281
length.class25-30	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	21,717	1,795
length.class25-30	p: ~length.class + ..time	0.00	10,614	952
length.class25-30	p: ~-1 + ..time	0.00	16,317	1,219
	Model averaged		14,519	1,560
length.class30-35	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	9,097	702
length.class30-35	p: ~-1 + length.class:..time	0.38	8,678	1,589
length.class30-35	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	7,685	728
length.class30-35	p: ~length.class + ..time	0.00	9,897	1,775
length.class30-35	p: ~-1 + ..time	0.00	8,898	681
	Model averaged		8,937	1,144
length.class35+	p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.62	4,662	383
length.class35+	p: ~-1 + length.class:..time	0.38	5,929	2,791
length.class35+	p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	2,075	271
length.class35+	p: ~length.class + ..time	0.00	13,244	6,369
length.class35+	p: ~-1 + ..time	0.00	4,278	345
	Model averaged		5,145	1,853

Notice how estimates of abundance are requested for each length class. Now the estimates of the sub-population abundances differ among the models even though the estimates overall population abundances are equal

7 Accounting for heterogeneity II - continuous fixed covariate

In the previous sections, we showed how a continuous covariate could be broken into a set of discrete classes and a stratified model applied. In some cases, it may be of interest to use a smooth function of the covariates (e.g., a quadratic curve, or a spline fit) to represent, for example, the catchability curve as a function of length.

This is relatively straight forward except for a few items that need to be addressed

• Standardizing the covariates to have a mean close to 0 and a standard deviation close to 1 improves numerical stability and convergence, especially when fitting a quadratic curve.
• Animals with a very low estimated catchability have a very large expansion factor when the Horvitz-Thompson estimator is formed. You may need to remove fish with very large expansion factors when estimating abundance.

7.1 Quadratic relationship between the probability of capture and length - Northern Pike

We again return to the Northern Pike example. We first standardize the length measurement (the ls variable in the revised data frame):

data(data_NorthernPike)

data_NorthernPike$ls <- scale(data_NorthernPike$length)
head(data_NorthernPike)
##   cap_hist length Sex freq         ls
## 1       01  23.20   M    1 -0.7146489
## 2       01  28.89   F    1  0.3710705
## 3       01  25.20   M    1 -0.3330252
## 4       01  22.20   M    1 -0.9054607
## 5       01  25.00   M    1 -0.3711876
## 6       01  24.70   M    1 -0.4284311

Now we fit several models and compare them to the previous models.

• a single quadratic curve (on the logit scale) with an additive sift between sampling events
• two separate quadratic curves (on the logit scale) for each sampling event

nop.fit.length.quad1 <- Petersen::LP_fit(data_NorthernPike,
                           p_model=~..time + ls + I(ls*2))
nop.fit.length.quad2 <- Petersen::LP_fit(data_NorthernPike, 
                           p_model=~..time + ls + I(ls^2) + 
                                    ..time:ls + ..time:I(ls^2))

We can compare models with the previous models using length classes using the usual AICc methods (Table 18).

# compare the various models

nop.sex.aictab <- LP_AICc(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2,
        nop.fit.length.quad1,
        nop.fit.length.quad2)  

Table 18: Comparison of models fit to the Northern Pike data including quadratic functions of length

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)	-3,575.331	6	7,805	7,162.67	0.00	1.00
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	-3,596.138	6	7,805	7,204.29	41.61	0.00
p: ~-1 + length.class:..time	-3,592.615	10	7,805	7,205.26	42.59	0.00
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	-3,621.220	6	7,805	7,254.45	91.78	0.00
p: ~length.class + ..time	-3,678.014	6	7,805	7,368.04	205.37	0.00
p: ~-1 + ..time	-3,702.341	2	7,805	7,408.68	246.01	0.00
p: ~..time + ls + I(ls * 2)	-3,702.290	4	7,805	7,412.59	249.91	0.00

The model with two separate quadratic fits is by far the best fitting model.

We obtain the estimate of abundance and look at the estimated relationship between the length and the probability of capture (Figure 11)

nop.est.length.quad2 <- Petersen::LP_est(nop.fit.length.quad2, N_hat=~1)

est.p <- nop.est.length.quad2$detail$data.expand
ggplot(data=est.p, aes(x=length, y=p, color=as.factor(..time)))+
   geom_point()+
   geom_line(size=.1)+
   scale_color_discrete(name="Sample\nevent")+
   xlab("Length (in)")+
   ylab("Estimated p(capture)")

Figure 11: Estimated probability of capture from quadratic relationship with length

We see that the relationship at time 1 is quite peaked but not so much for sampling event 2 which is in accordance to the results from classifying length into discrete bins.

This model also shows a positive association between catchability at the two sampling events (Figure 12)

est.p <- nop.est.length.quad2$detail$data.expand
temp <- tidyr::pivot_wider(est.p,
                           id_cols=c("..index", "length"),
                           values_from="p",
                           names_from="..time",
                           names_prefix="t")
ggplot(data=temp, aes(x=t1, y=t2, color=length))+
   geom_point()+
   scale_color_continuous(name="Length")+
   xlab("Estimated p(capture) at sample event 1")+
   ylab("Estimated p(capture) at sample event 2")

Figure 12: Estimated relationship between catchability at two sampling events

The odd shape to the relationship is an artefact of the quadratic curves where the top/bottom half of the “curve” correspond to the ascending/descending limbs of the quadratic curve seen in Figure 11.

Figure 12 implies a positive correlation between the two capture probabilities which implies that the pooled-Petersen estimator will have a negative bias as seen in the model averaged results (Table 20).

The estimated abundance from the best fitting model is:

Table 19: Estimated abundance of Northern Pike from quadratic relationship with length

nop.est.length.quad2$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 59209.91 9365.551             0.95              logN
##   N_hat_LCL N_hat_UCL                                             p_model
## 1  43426.54  80729.73 ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)
##                                               name_model   cond.ll n.parms nobs
## 1 p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2) -3575.331       6 7805
##    method
## 1 CondLik

The estimate appears on the large side with a large standard error compared to the other estimates seen previously. We suspect that some histories have a very small probability of capture and a large expansion factor as shown in Figure 13

est.ef <- nop.est.length.quad2$detail$data
ggplot(data=est.ef, aes(x=length, y=..EF))+
   geom_point()+
   geom_line()+
   xlab("Length (in)")+
   ylab("Estimated expansion factor")

Figure 13: Estimated expansion factor by length from quadratic relationship with length

We can see that there are several large fish with expansion factors that appear to be much larger than the majority of fish. We can estimate the abundance truncating the expansion factor, say at 30:

nop.est.length.quad2.tef <- Petersen::LP_est(nop.fit.length.quad2, 
                                N_hat=~-1+I(as.numeric(..EF<30)))

plyr::rbind.fill(nop.est.length.quad2    $summary,
                 nop.est.length.quad2.tef$summary)
##                          N_hat_f                 N_hat_rn    N_hat N_hat_SE
## 1                             ~1              (Intercept) 59209.91 9365.551
## 2 ~-1 + I(as.numeric(..EF < 30)) I(as.numeric(..EF < 30)) 58866.46 9005.852
##   N_hat_conf_level N_hat_conf_method N_hat_LCL N_hat_UCL
## 1             0.95              logN  43426.54  80729.73
## 2             0.95              logN  43615.86  79449.54
##                                               p_model
## 1 ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)
## 2 ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)
##                                               name_model   cond.ll n.parms nobs
## 1 p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2) -3575.331       6 7805
## 2 p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2) -3575.331       6 7805
##    method
## 1 CondLik
## 2 CondLik

Notice the use of the special variable ..EF in the model for N_hat. The estimate is reduced (as expected) but not by a great extent, so there is no real reason to adopt this second estimate.

Finally, we can again do the model averaging (Table 20):

# extract the estimates of the abundance 
nop.length.class.ma.N_hat_length.class<- LP_modavg(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, 
        nop.fit.length.quad1,
        nop.fit.length.quad2,   N_hat=~1)  

Table 20: Model averaged estimates of abundance including the quadratic fit to length

Modnames	AICcWt	Estimate	SE
p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)	1.00	59,210	9,366
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.00	49,536	3,629
p: ~-1 + length.class:..time	0.00	60,614	13,039
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	49,536	3,629
p: ~length.class + ..time	0.00	93,218	39,716
p: ~-1 + ..time	0.00	49,536	3,629
p: ~..time + ls + I(ls * 2)	0.00	49,563	3,635
Model averaged		59,210	9,366

The quadratic model is so much a better fit that it overwhelms the other models and the model averaged estimate is basically the first model. Notice that the model averaged confidence interval is larger than the original model because the variation in estimates among the models has also been taken into account.

It would be interesting to fit quadratic on log(length) to give a skewed catchability curve.

7.2 Spline relationship between the probability of capture and length - Northern Pike

The quadratic curve fairly ridged in its shape. An alternate way to fit a curve to the catchabilities is through the use of splines.

Give a tutorial on splines here and how to choose the appropriate values for df etc - use AIC to select

We continue with the Northern Pike example and fit a model with a separate spline curve at each sample event.

nop.fit.length.sp2.df4<- Petersen::LP_fit(data_NorthernPike, 
                            p_model=~..time+bs(ls, df=4) + 
                                     ..time:bs(ls, df=4))

nop.fit.length.sp2.df5<- Petersen::LP_fit(data_NorthernPike, 
                            p_model=~..time+bs(ls, df=5) +
                                     ..time:bs(ls, df=5))

We can compare models using the usual AICc methods (Table 21).

# compare the various models

nop.sex.aictab <- LP_AICc(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2,
        nop.fit.length.quad1,
        nop.fit.length.quad2,
        nop.fit.length.sp2.df4, nop.fit.length.sp2.df5)  

Table 21: Comparison of models fit to the Northern Pike data including spline functions of length

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~..time + bs(ls, df = 4) + ..time:bs(ls, df = 4)	-3,565.507	10	7,805	7,151.04	0.00	0.85
p: ~..time + bs(ls, df = 5) + ..time:bs(ls, df = 5)	-3,565.290	12	7,805	7,154.62	3.58	0.14
p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)	-3,575.331	6	7,805	7,162.67	11.63	0.00
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	-3,596.138	6	7,805	7,204.29	53.24	0.00
p: ~-1 + length.class:..time	-3,592.615	10	7,805	7,205.26	54.22	0.00
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	-3,621.220	6	7,805	7,254.45	103.41	0.00
p: ~length.class + ..time	-3,678.014	6	7,805	7,368.04	217.00	0.00
p: ~-1 + ..time	-3,702.341	2	7,805	7,408.68	257.64	0.00
p: ~..time + ls + I(ls * 2)	-3,702.290	4	7,805	7,412.59	261.54	0.00

The model with two separate splines and 4 df is again the best model in the set. We obtain the estimate of abundance and look at the estimated relationship between the length and the probability of capture (Figure 14)

nop.est.length.sp2.df4 <- Petersen::LP_est(nop.fit.length.sp2.df4, N_hat=~1)

est.p <- nop.est.length.sp2.df4$detail$data.expand
ggplot(data=est.p, aes(x=length, y=p, color=as.factor(..time)))+
   geom_point()+
   geom_line(size=.1)+
   scale_color_discrete(name="Sample\nevent")+
   xlab("Length (in)")+
   ylab("Estimated p(capture)")

Figure 14: Estimated probability of capture from spline fit with length

We see that the relationship at time 1 is quite peaked but not so much for sampling event 2 which is in accordance to the results from classifying length into discrete bins. This spline model seems to indicate a very high catchability of very large fish at the first sampling event, something that was missed when the simple quadratic curve was used.

This model also shows a positive association between catchability at the two sampling events (Figure 15)

est.p <- nop.est.length.sp2.df4$detail$data.expand
temp <- tidyr::pivot_wider(est.p,
                           id_cols=c("..index", "length"),
                           values_from="p",
                           names_from="..time",
                           names_prefix="t")
ggplot(data=temp, aes(x=t1, y=t2, color=length))+
   geom_point()+
   scale_color_continuous(name="Length")+
   xlab("Estimated p(capture) at sample event 1")+
   ylab("Estimated p(capture) at sample event 2")

Figure 15: Estimated relationship between catchability at two sampling events from spline fit

The odd shape to the relationship is an artefact of two spline curves and the differing relationship between the ascending/descending arms of the spline. The above fit implies a positive correlation between the two capture probabilities which implies that the pooled-Petersen estimator will have a negative bias as seen in the model averaged results (Table 22).

nop.est.length.sp2.df4$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 60485.41 12635.26             0.95              logN
##   N_hat_LCL N_hat_UCL                                          p_model
## 1  40163.97  91088.72 ~..time + bs(ls, df = 4) + ..time:bs(ls, df = 4)
##                                            name_model   cond.ll n.parms nobs
## 1 p: ~..time + bs(ls, df = 4) + ..time:bs(ls, df = 4) -3565.507      10 7805
##    method
## 1 CondLik

Finally, we can again do the model averaging (Table 22):

# extract the estimates of the abundance 
nop.length.class.ma.N_hat_length.class<- LP_modavg(
        nop.fit.time,
        nop.fit.length.class.time,
        nop.fit.length.class.p.time,
        nop.fit.eq.t1,
        nop.fit.eq.t2, 
        nop.fit.length.quad1,
        nop.fit.length.quad2,
        nop.fit.length.sp2.df4, nop.fit.length.sp2.df5,  N_hat=~1)  

Table 22: Model averaged estimates of abundance including the spline fit to length

Modnames	AICcWt	Estimate	SE
p: ~..time + bs(ls, df = 4) + ..time:bs(ls, df = 4)	0.85	60,485	12,635
p: ~..time + bs(ls, df = 5) + ..time:bs(ls, df = 5)	0.14	59,567	10,895
p: ~..time + ls + I(ls^2) + ..time:ls + ..time:I(ls^2)	0.00	59,210	9,366
p: ~-1 + I(as.numeric(..time == 1)) + I(as.numeric(..time == 2)):length.class	0.00	49,536	3,629
p: ~-1 + length.class:..time	0.00	60,614	13,039
p: ~-1 + I(as.numeric(..time == 2)) + I(as.numeric(..time == 1)):length.class	0.00	49,536	3,629
p: ~length.class + ..time	0.00	93,218	39,716
p: ~-1 + ..time	0.00	49,536	3,629
p: ~..time + ls + I(ls * 2)	0.00	49,563	3,635
Model averaged		60,351	12,398

The spline model is so much a better fit that it overwhelms the other models and the model averaged estimate is basically the first model. Notice that the model averaged confidence interval is larger than the original model because the variation in estimates among the models has also been taken into account.

The use of a spline fit is an alternative to the methods of Chen and LLoyd (2000) who used a non-parametric smoother to estimate the catchabilities at each sampling event. The advantage of the spline method is that the AICc framework can be used to rank the various models.

7.3 Non-parametric smoothing

Chen and Lloyd (2000) developed a non-parametric smoother for the relationship between a continuous covariate and the catchability at the sample events. The data are divided into bins, a stratified-Petersen estimator is used on each bin, and a smoother is used to avoid the capture probabilities or the abundance estimates from varying wildly between successive bins.

The default fit is obtained using:

# fit the Chen and Lloyd estimator using default bin width
data(data_NorthernPike)
nop.CL <- Petersen::LP_CL_fit(data_NorthernPike, covariate="length")

A plot of the recapture probabilities shows the upwards recapture probabilities with the larger lengths (Figure 16)

nop.CL$fit$plot1

Figure 16: Estimated probabilities of recapture by length

And a plot of the abundance as a function of length along with the estimated overall abundance (Figure 17)

nop.CL$fit$plot2
nop.CL$summary
##   N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 56620.94 5509.565             0.95              logN
##   N_hat_LCL N_hat_UCL p_model cond.ll n.parms nobs    method
## 1  46789.66  68517.92      NA      NA      NA 7805 ChenLloyd

Figure 17: Estimated abundance of Northern Pike by length

The estimated abundance is comparable to the estimates from the using the semi-parametric spline method with a smaller standard error.

Because this is a non-parametric fit, it is not possible to compute a likelihood value and so AICc methods can not be used to compare this model with the previous models. However, the spline fit has a comparable flexibility and fits nicely into the AICc framework.

More bins leads to a “wigglier fit” and some playing with the smoothing parameters is needed to avoid spikes in abundance which are artefacts of the small sample sizes (especially recaptures) in the smaller bins seen around lengths of 40 inches or higher. Generally speaking, the smaller the bins, the larger the smoothing standard deviation and vice-verso.

# fit the Chen and Lloyd estimator with lower smoothing parameter
data(data_NorthernPike)
old.centers <- nop.CL$covar.data$center
nop.CL2 <- Petersen::LP_CL_fit(data_NorthernPike, covariate="length",
                           centers=seq(17, 43,1), h1=1.5, h2=1.5)
nop.CL2$fit$plot1

nop.CL2$fit$plot2

nop.CL2$summary
##   N_hat_f    N_hat_rn   N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
## 1      ~1 (Intercept) 56982.7 5642.788             0.95              logN
##   N_hat_LCL N_hat_UCL p_model cond.ll n.parms nobs    method
## 1  46930.12  69188.58      NA      NA      NA 7805 ChenLloyd

8 Accounting for heterogeneity III - geographic or temporal stratification

8.1 Introduction

In some cases, heterogeneity in catchability at both sampling events is related to the location where the fish was sampled (i.e., geographical stratification) or the date when the fish was sampled (temporal stratification) or both.

Data consists of the number of fish releases in each of \(s\) release strata (\(n_{1s}\)), which are recaptured in one of \(t\) recapture strata, leading to an \(s \times t\) recapture matrix (\(m_{ij}\)). Finally, there are \(n_{2t}\) unmarked fish captured for the first time in the second sampling event in the \(t\) recapture strata.

The key difference between these two cases is that in geographic stratification, it is theoretically possible to move from any one geographic stratum to any other geographic stratum leading to a general movement matrix. However, in temporal stratification, you cannot capture the fish at the second sampling event before it is released, leading to recapture matrices that have zeros on the lower diagonal.

Schaefer (1951) developed an estimate of the total population abundance, \(N\), using ratio and expectation arguments. The Schaefer estimate is biased unless capture probabilities are equal in all initial strata or the recovery probabilities are the same in all the final strata. If either condition holds, the pooled Petersen will also be unbiased and will be more precise (it makes more efficient use of the data). No estimate of standard error (s.e.) is available for the Schaefer estimate. Consequently, the use of the Schaefer (1951) estimator is no longer recommended

Seber (1982) summarizes the early work of Chapman and Junge (1956) and Darroch (1961). There are 3 cases:(a) \(s=t\); (b) \(s<t\); and (c) \(s>t\). In case (a), the recapture matrix (\(m\)) is square, and a simple matrix-based estimator is available. Both the initial and final stratum (population) abundances can be estimated. This estimate is commonly referred to as the Darroch estimate. They gave the necessary and sufficient conditions for the pooled Petersen to be unbiased and developed two chi-square tests for sufficient conditions (i.e., if the tests fail to detect an effect, it may “safe” to pool…if tests detect an effect, it may or may not be safe to pool).These are the two \(\chi^2\) tests presented earlier.

In case (b), only the initial stratum (population) abundances can be estimated, and in case (c) only the final stratum abundances can be estimated. The total population size, \(N\), is nevertheless estimable. Plante (1990) developed an alternate maximum likelihood method for the Darroch estimate that can be applied to all 3 cases including the cases where \(s \ne t\).

A key issue with geographic stratification, is that it is very sensitive to singularities in the recapture matrix. For example, it fails if any row is a multiple of other rows, or a linear combination of other rows. It leads to estimates with very large standard errors (and nonsensical stratum estimates) if the recapture matrix is close to singularity.

This often requires pooling of rows or columns to reduce the singularity of the recapture matrix. Schwarz and Taylor (1998) review the estimators for geographic stratification and provides guidance on how to pool rows or columns. This data are also often sparse requiring much pooling. This is difficult to do in a structured fashion. If all rows are pooled to a single row, or all columns pooled to a single column, the estimator reduces to a Pooled-Petersen estimator.

Arnason et al (1996) created a Windows program (SPAS) to help in the analysis of geographical stratification. Schwarz (2023) ported the functionality to R and included “logical pooling” so that different poolings could be compared using AIC etc. The Petersen package provides a wrapper to make it easier to use. The SPAS software also includes an autopool() function.

While SPAS could also be used for temporal stratification, but the problem is that there is no way to use the “structural zeros” representing fish being captured before being released, or fish being recaptured more than a small number of weeks after being released. This gives a diagonal or band-diagonal structure to the temporally stratified data which SPAS ignores.

Bjorkstedt (2000). created the DARR (Darroch Analysis with Rank-Reduction) computer program to automatically pool the above sparse matrices but this program uses simple rules to ensure that adequate sample sizes are available in each of the release and recovery strata. It ignores the highly structured form of the data and cannot deal with many problems commonly encountered in such data such as missing strata. In response to this, Bonner and Schwarz (2011) developed BTSPAS which uses a Bayesian approach with a hierarchical model for the capture probabilities to share information when data are spares, and a spline for the run shape to again share information and to provide a straightforward way to interpolate over missing data. Again, a wrapper is provided in the Petersen package to simplify usage of BTSPAS.

8.2 Geographic stratification

8.2.1 Sampling protocol

Consider an study to estimate the number of return salmon to a river. The returns extends over several weeks and there are several spawning sites. As the adult salmon return, they are captured and marked with individually numbered tags and released at the first capture location using, for example, a fishwheel. The migration continues, and stream walks of the spawning grounds notes the number of tagged and untagged fish..

The efficiency of the fishwheels varies over time in response to stream flow, run size passing the wheel and other uncontrollable events. So it is unlikely that the capture probabilities are equal over time, i.e. are heterogeneous over time. Similarly, the effort at the spawning grounds varies by ground and is also heterogeneous.

This is a form of temporal x geographical stratification – the key feature in geographical stratification is that fish from any of the tagging strata, can, in theory, go to any of the recovery strata. There is no “natural” ordering of the geographical strata; the temporal strata have a natural ordering.

In the more general case, both strata are “geographically stratified”.

8.2.2 Data structure

The same data structure as previously seen is also used, except the capture history is modified to account for potentially many geographical or temporal strata as follows:

• xx..yy represents a capture_history where xx and yy are the temporal or geographical stratum (e.g., julian week or spawning area) and ‘..’ separates the two strata from the two sampling event. If a fish is released in a stratum xx and never captured again, then yy is set to 0; if a fish is newly captured in stratum yy, then xx is set to zero.
• frequency variable for the number of fish with this capture history
• other covariates for the stratum yy.

It is not possible to model the initial capture probability or to separate fish by additional stratification variables in the SPAS software. These have rarely found to be useful in the geographically stratified models.

Some care is needed to properly account for 0 or missing values. Strata with 0 releases, should be dropped from the data structure. Similarly, strata with 0 recapture and 0 newly tagged fish should also be dropped. As shown later, some pooling of sparse rows/columns may be needed.

8.2.3 Key assumptions

The new key assumptions for this method are:

• all marked fish move to the recovery strata in the same proportions as unmarked fish in any release stratum
• there is no fall back of tagged fish, i.e., after marking the marked fish continue on their migration in same fashion as unmarked fish.
• catchability of marked fish represents the catchability of the unmarked fish

Unfortunately, for most of these assumptions, there is little information in the data that help detect violations. Gross violations of this model can be detected from the goodness-of-fit tests presented in Arnason et al. (1996) and the output from SPAS when the number of rows is not equal to the number of columns.

8.2.4 Harrison River Example

Returning salmon were captured and tags at a down river trap. A colored tag that varied by week of capture was applied. During the spawning season, stream walks took place at several spawning sites where the number of tags by color and the number of untagged fish was recorded on a weekly basis.

This is a combination of temporal stratification (when tags applied) and geographic stratification the spawning areas.

Here is part of the data:

data(data_spas_harrison)
head(data_spas_harrison, n=10)

   cap_hist freq
1    01..00  130
2    02..00  330
3    03..00  790
4    04..00  667
5    05..00  309
6    06..00   65
7     00..a  744
8     01..a    4
9     02..a   12
10    03..a    7

Because we allow the stratum labels to be longer than a single digit, we use the “..” to separate strata labels in the capture history.

In week 1, 130 fish were tagged and never seen again; 4 fish were tagged in week 1 and recovered in area a, etc. This data can be summarized into a matrix:

    fw2
fw1    00    a    b    c    d    e    f
  00    0  744 1187 2136  951  608  127
  01  130    4    2    1    1    0    0
  02  330   12    7   14    1    3    0
  03  790    7   11   41    9    1    1
  04  667    1   13   40   12    9    1
  05  309    0    1    8    8    3    0
  06   65    0    0    0    0    0    1

The first row (corresponding to fw1=0) are the number of unmarked fish recovered in spawning area a. The first column (corresponding to fw2=0) are the number of released marked fish that were never seen again. The matrix in the lower bottom of the above array shows the number of fish released in each temporal stratum and recaptured in each spawning area.

8.2.5 Fitting the SPAS model

The SPAS model is fit to the above data using the wrapper supplied with this package. If a finer control is wanted on the fit, please refer to the documentation from the SPAS package. The very sparse last row will make fitting the model to the original data difficult with a very flat likelihood surface and so we relax the convergence criteria:

The row.pool.in and col.pool.in inform the function on which rows or columns to pool before the analysis proceeds. Both parameters use a vector of codes (length \(s\) for row pooing and length \(t\) for column pooling) where rows/columns are pooled that share the same value.

For example. row.pool.in=c(1,2,3,4,5,6) would imply that no rows are pooled, while row.pool.in=c('a','a','a','b','b','b') would imply that the first three rows and last three rows are pooled. The entries in the vector can be numeric or character; however, using character entries implies that the final pooled matrix is displayed in the order of the character entries. I find that using entries such as '123' to represent pooling rows 1, 2, and 3 to be easiest to use.

The SPAS system only fits models were the number of rows after pooling is less than or equal to the number of columns after pooling.

8.2.6 Results from model 1 (no pooling).

The summary of the fit is:

mod..1$summary

                     p_model      name_model  cond.ll n.parms nobs method
1 Refer to row.pool/col.pool No restrictions 47243.45      48 8256   SPAS
  cond.factor
1    4121.288

The usual conditional likelihood is presented etc. Of more importance is the condition factor. This indicated how close to singularity the recovery matrix is with larger values indicating closer to singularity. Usually, this value should be 1000 or less to avoid numerical issues in the fit.

The results of the model fit is a LARGE list. But the SPAS.print.model() function produces a nice report

SPAS::SPAS.print.model(mod..1$fit)

Model Name: No restrictions 
   Date of Fit: 2025-10-24 15:54 
   Version of OPEN SPAS used : SPAS-R 2025.2.1 
 
Raw data 
     a    b    c   d   e   f    
01   4    2    1   1   0   0 130
02  12    7   14   1   3   0 330
03   7   11   41   9   1   1 790
04   1   13   40  12   9   1 667
05   0    1    8   8   3   0 309
06   0    0    0   0   0   1  65
   744 1187 2136 951 608 127   0

Row pooling setup : 1 2 3 4 5 6 
Col pooling setup : 1 2 3 4 5 6 
Physical pooling  : FALSE 
Theta pooling     : FALSE 
CJS pooling       : FALSE 


Chapman estimator of population size  70135  (SE  4503  )
 

Raw data AFTER PHYSICAL (but not logical) POOLING 
       pool1 pool2 pool3 pool4 pool5 pool6    
pool.1     4     2     1     1     0     0 130
pool.2    12     7    14     1     3     0 330
pool.3     7    11    41     9     1     1 790
pool.4     1    13    40    12     9     1 667
pool.5     0     1     8     8     3     0 309
pool.6     0     0     0     0     0     1  65
         744  1187  2136   951   608   127   0

Condition number of XX' where X= (physically) pooled matrix is  4121.288 
Condition number of XX' after logical pooling                   4121.288 

Large value of kappa (>1000) indicate that rows are approximately proportional which is not good

  Conditional   Log-Likelihood: 47243.45    ;  np: 48 ;  AICc: -94390.9 

  Code/Message from optimization is:  0 relative convergence (4) 

Estimates
             pool1  pool2  pool3 pool4 pool5 pool6 psi cap.prob exp factor
pool.1         3.7    2.6    0.8   0.9   0.0     0 130    0.005      191.2
pool.2        12.0    7.0   14.0   1.0   3.0     0 330    1.000        0.0
pool.3         7.0   11.0   41.0   9.0   1.0     1 790    1.000        0.0
pool.4         1.0   13.8   37.5  11.8  10.9     1 667    0.021       47.6
pool.5         0.0    1.0    7.7   7.9   3.3     0 309    0.037       26.3
pool.6         0.0    0.0    0.0   0.0   0.0     1  65    0.012       79.4
est unmarked 744.0 1186.0 2139.0 951.0 606.0   127   0       NA         NA
             Pop Est
pool.1         26523
pool.2           367
pool.3           860
pool.4         36104
pool.5          8998
pool.6          5307
est unmarked   78159

SE of above estimates
             pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
pool.1         1.6   1.3   0.8   1.0   0.0     0 11.4    0.002       83.1
pool.2         3.5   2.6   3.7   1.0   1.7     0 18.2    0.000        0.0
pool.3         2.6   3.3   6.4   3.0   1.0     1 28.1    0.000        0.0
pool.4         1.0   3.6   5.9   3.5   2.1     1 25.8    0.007       16.7
pool.5         0.0   1.0   2.7   2.9   2.0     0 17.6    0.072       53.7
pool.6         0.0   0.0   0.0   0.0   0.0     1  8.1    0.015       94.7
est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
             Pop Est
pool.1         11462
pool.2             0
pool.3             0
pool.4         12411
pool.5         17661
pool.6          6253
est unmarked   14676


Chisquare gof cutoff  : 0.1 
Chisquare gof value   : 0.843434 
Chisquare gof df      : 0 
Chisquare gof p       : NA 

The original data, the data after pooling, estimates and their standard errors are shown. Here the stratified-Petersen estimate of the total number of fish passing the first sampling station is 78,159 with a standard error of 14,676.

The N entries refer to the population size; the N.stratum entries refer to the individual stratum population sizes; the cap entries refer to the estimated probability of capture in each row stratum; the exp.factor entries refer to (1-cap)/cap, or the expansion factor for each row; the psi entries refer to the number of animals tagged but never seen again (the right most column in the input data); and the theta entries refer to the expected number of animals that were tagged in row stratum \(i\) and recovered in column stratum \(j\) (after pooling).

The fit is not entirely satisfactory because notice the very small population estimates for in weeks 2 and 3 or releases. This is unrealistic and is an indication of potential singularity problems in the data as noted with the large condition number seen earlier.

You can extract parts of the “printed” output by using the extract=TRUE argument in the SPAS.print.model() function:

# demonstrate how to extract part of output
mod..1.res <- SPAS::SPAS.print.model(mod..1$fit, extract=TRUE)
names(mod..1.res)

[1] "model_name" "date"       "version"    "input"      "Chapman"   
[6] "fit"        "spas"       "gof"       

mod..1.res$spas$estimate

             pool1  pool2  pool3 pool4 pool5 pool6 psi cap.prob exp factor
pool.1         3.7    2.6    0.8   0.9   0.0     0 130    0.005      191.2
pool.2        12.0    7.0   14.0   1.0   3.0     0 330    1.000        0.0
pool.3         7.0   11.0   41.0   9.0   1.0     1 790    1.000        0.0
pool.4         1.0   13.8   37.5  11.8  10.9     1 667    0.021       47.6
pool.5         0.0    1.0    7.7   7.9   3.3     0 309    0.037       26.3
pool.6         0.0    0.0    0.0   0.0   0.0     1  65    0.012       79.4
est unmarked 744.0 1186.0 2139.0 951.0 606.0   127   0       NA         NA
             Pop Est
pool.1         26523
pool.2           367
pool.3           860
pool.4         36104
pool.5          8998
pool.6          5307
est unmarked   78159

mod..1.res$spas$se

             pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
pool.1         1.6   1.3   0.8   1.0   0.0     0 11.4    0.002       83.1
pool.2         3.5   2.6   3.7   1.0   1.7     0 18.2    0.000        0.0
pool.3         2.6   3.3   6.4   3.0   1.0     1 28.1    0.000        0.0
pool.4         1.0   3.6   5.9   3.5   2.1     1 25.8    0.007       16.7
pool.5         0.0   1.0   2.7   2.9   2.0     0 17.6    0.072       53.7
pool.6         0.0   0.0   0.0   0.0   0.0     1  8.1    0.015       94.7
est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
             Pop Est
pool.1         11462
pool.2             0
pool.3             0
pool.4         12411
pool.5         17661
pool.6          6253
est unmarked   14676

cat("\n\nEstimated total abundance ", mod..1.res$spas$estimate[nrow(mod..1.res$spas$estimate), ncol(mod..1.res$spas$estimate)],
    "(SE ", mod..1.res$spas$se[nrow(mod..1.res$spas$estimate), ncol(mod..1.res$spas$estimate)], ") fish", "\n")

Estimated total abundance  78159 (SE  14676 ) fish 

8.2.7 Pooling some rows and columns

As noted by Darroch (1961), the stratified-Petersen will fail if the matrix of movements is close to singular. This often happens if two rows are proportional to each other. In this case, there is no unique MLE for the probability of capture in the two rows, and they should be pooled. A detailed discussion of pooling is found in Schwarz and Taylor (1998).

There is no simple way to determine which rows/column to pool but when estimated population sizes are small for a stratum, this is usual an indication that some pooling is required. The SPAS system has an experimental autopool feature which may serve as a guide.

8.2.7.1 Pooling pairs of rows

Let us now pool the first two rows, the next two rows, and the last two rows,

The code is

mod..2 <- Petersen::LP_SPAS_fit(data_spas_harrison, model.id="Pooling every second row",
                       row.pool.in=c("12","12","34","44","56","56"),
                       col.pool.in=c(1,2,3,4,5,6))

Using nlminb to find conditional MLE
outer mgc:  1967.318 
outer mgc:  1585.784 
outer mgc:  601.797 
outer mgc:  361.6262 
outer mgc:  255.3465 
outer mgc:  301.4938 
outer mgc:  25.14831 
outer mgc:  2.82591 
outer mgc:  2.424709 
outer mgc:  16.30222 
outer mgc:  4.565925 
outer mgc:  24.77543 
outer mgc:  57.44612 
outer mgc:  7.92634 
outer mgc:  11.63666 
outer mgc:  10.76265 
outer mgc:  3.863489 
outer mgc:  13.76323 
outer mgc:  26.44275 
outer mgc:  4.706843 
outer mgc:  23.51753 
outer mgc:  0.8778394 
outer mgc:  5.614518 
outer mgc:  1.738815 
outer mgc:  5.426214 
outer mgc:  16.44758 
outer mgc:  1.748598 
outer mgc:  1.453996 
outer mgc:  6.546109 
outer mgc:  6.563646 
outer mgc:  8.584827 
outer mgc:  20.45095 
outer mgc:  2.996709 
outer mgc:  1.732739 
outer mgc:  5.660383 
outer mgc:  6.150282 
outer mgc:  6.135996 
outer mgc:  10.84269 
outer mgc:  2.233833 
outer mgc:  5.003054 
outer mgc:  2.24289 
outer mgc:  2.801219 
outer mgc:  0.6797359 
outer mgc:  0.7992463 
outer mgc:  0.06792093 
outer mgc:  0.1106809 
outer mgc:  0.04946835 
outer mgc:  0.02579226 
outer mgc:  0.01303348 
outer mgc:  0.006503766 
outer mgc:  0.003219488 
outer mgc:  0.001356174 
outer mgc:  0.001004277 
outer mgc:  0.0005343734 
outer mgc:  0.00013112 
Convergence codes from nlminb  1 singular convergence (7) 
Finding conditional estimate of N

#mod..2$summary

Notice how we specify the pooling for rows and columns and the choice of entries for the two corresponding vectors. Now the condition factor is much better (well less than 1000).

The results of the model fit is

Model Name: Pooling every second row 
   Date of Fit: 2025-10-24 15:54 
   Version of OPEN SPAS used : SPAS-R 2025.2.1 
 
Raw data 
     a    b    c   d   e   f    
01   4    2    1   1   0   0 130
02  12    7   14   1   3   0 330
03   7   11   41   9   1   1 790
04   1   13   40  12   9   1 667
05   0    1    8   8   3   0 309
06   0    0    0   0   0   1  65
   744 1187 2136 951 608 127   0

Row pooling setup : 12 12 34 44 56 56 
Col pooling setup : 1 2 3 4 5 6 
Physical pooling  : FALSE 
Theta pooling     : FALSE 
CJS pooling       : FALSE 


Chapman estimator of population size  70135  (SE  4503  )
 

Raw data AFTER PHYSICAL (but not logical) POOLING 
        pool1 pool2 pool3 pool4 pool5 pool6    
pool.12     4     2     1     1     0     0 130
pool.12    12     7    14     1     3     0 330
pool.34     7    11    41     9     1     1 790
pool.44     1    13    40    12     9     1 667
pool.56     0     1     8     8     3     0 309
pool.56     0     0     0     0     0     1  65
          744  1187  2136   951   608   127   0

Condition number of XX' where X= (physically) pooled matrix is  4121.288 
Condition number of XX' after logical pooling                   189.4612 

Large value of kappa (>1000) indicate that rows are approximately proportional which is not good

  Conditional   Log-Likelihood: 47242.37    ;  np: 46 ;  AICc: -94392.74 

  Code/Message from optimization is:  1 singular convergence (7) 

Estimates
             pool1  pool2  pool3 pool4 pool5 pool6 psi cap.prob exp factor
pool.12        3.5    2.8    0.9     1   0.0   0.0 130    0.019       52.1
pool.12       10.4   10.0   12.4     1   3.1   0.0 330    0.019       52.1
pool.34        7.0   11.0   41.0     9   1.0   1.0 790    1.000        0.0
pool.44        0.9   15.3   37.5    12   9.1   1.1 667    0.036       26.6
pool.56        0.0    1.6    6.9     8   3.1   0.0 309    0.015       66.2
pool.56        0.0    0.0    0.0     0   0.0   1.5  65    0.015       66.2
est unmarked 746.0 1180.0 2141.0   951 608.0 126.0   0       NA         NA
             Pop Est
pool.12         7327
pool.12        19484
pool.34          860
pool.44        20470
pool.56        22123
pool.56         4438
est unmarked   74701

SE of above estimates
             pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
pool.12        1.8   1.9   0.9   1.0   0.0   0.0 11.4    0.006       15.6
pool.12        3.3   3.0   3.2   1.0   1.5   0.0 18.2    0.006       15.6
pool.34        2.6   3.3   6.4   3.0   1.0   1.0 28.1    0.000        0.0
pool.44        0.9   4.4   6.6   3.4   2.7   1.1 25.8    0.021       16.0
pool.56        0.0   1.7   2.3   2.6   1.4   0.0 17.6    0.009       39.5
pool.56        0.0   0.0   0.0   0.0   0.0   0.7  8.1    0.009       39.5
est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
             Pop Est
pool.12         2149
pool.12         5715
pool.34            0
pool.44        11887
pool.56        13007
pool.56         2609
est unmarked   10284


Chisquare gof cutoff  : 0.1 
Chisquare gof value   : 2.857233 
Chisquare gof df      : 2 
Chisquare gof p       : 0.2396402 

Here the stratified-Petersen estimate of the total number of smolts passing the first sampling station is 74,701 with a standard error of 74,701. which is a slight reduction from the unpooled estimates.

The estimates look better, but a standard error of 0 for the abundance in some strata is an indication that the fit is not very realistic.

8.2.7.2 Pooling to early vs late.

Let us now pool the first three rows and the last three rows,

The code is

mod..3 <- Petersen::LP_SPAS_fit(data_spas_harrison, model.id="Pooling early vs. late",
                       row.pool.in=c("123","123","123","456","456","456"),
                       col.pool.in=c(1,2,3,4,5,6))

Using nlminb to find conditional MLE
outer mgc:  2986.733 
outer mgc:  1769.526 
outer mgc:  816.8674 
outer mgc:  269.1311 
outer mgc:  30.10549 
outer mgc:  24.07398 
outer mgc:  5.393645 
outer mgc:  18.96964 
outer mgc:  2.953744 
outer mgc:  1.395084 
outer mgc:  18.87373 
outer mgc:  6.688677 
outer mgc:  1.640811 
outer mgc:  7.332616 
outer mgc:  1.685392 
outer mgc:  7.317009 
outer mgc:  1.522422 
outer mgc:  6.294555 
outer mgc:  1.261084 
outer mgc:  4.980596 
outer mgc:  2.103006 
outer mgc:  8.561154 
outer mgc:  1.79518 
outer mgc:  6.70074 
outer mgc:  2.203129 
outer mgc:  8.175411 
outer mgc:  2.210091 
outer mgc:  7.714625 
outer mgc:  3.180497 
outer mgc:  11.01373 
outer mgc:  4.797201 
outer mgc:  18.68681 
outer mgc:  10.12453 
outer mgc:  12.11322 
outer mgc:  6.303981 
outer mgc:  6.197803 
outer mgc:  2.248121 
outer mgc:  0.7088488 
outer mgc:  0.5721706 
outer mgc:  0.1888875 
outer mgc:  0.07641965 
outer mgc:  0.02903072 
outer mgc:  0.01084322 
outer mgc:  0.004013247 
outer mgc:  0.001470131 
outer mgc:  0.0001335366 
outer mgc:  9.781857e-06 
Convergence codes from nlminb  1 singular convergence (7) 
Finding conditional estimate of N

#mod..3$summary

Notice how we specify the pooling for rows and columns and the choice of entries for the two corresponding vectors. Now the condition factor is again much better (well less than 1000).

The results of the model fit is

Model Name: Pooling early vs. late 
   Date of Fit: 2025-10-24 15:54 
   Version of OPEN SPAS used : SPAS-R 2025.2.1 
 
Raw data 
     a    b    c   d   e   f    
01   4    2    1   1   0   0 130
02  12    7   14   1   3   0 330
03   7   11   41   9   1   1 790
04   1   13   40  12   9   1 667
05   0    1    8   8   3   0 309
06   0    0    0   0   0   1  65
   744 1187 2136 951 608 127   0

Row pooling setup : 123 123 123 456 456 456 
Col pooling setup : 1 2 3 4 5 6 
Physical pooling  : FALSE 
Theta pooling     : FALSE 
CJS pooling       : FALSE 


Chapman estimator of population size  70135  (SE  4503  )
 

Raw data AFTER PHYSICAL (but not logical) POOLING 
         pool1 pool2 pool3 pool4 pool5 pool6    
pool.123     4     2     1     1     0     0 130
pool.123    12     7    14     1     3     0 330
pool.123     7    11    41     9     1     1 790
pool.456     1    13    40    12     9     1 667
pool.456     0     1     8     8     3     0 309
pool.456     0     0     0     0     0     1  65
           744  1187  2136   951   608   127   0

Condition number of XX' where X= (physically) pooled matrix is  4121.288 
Condition number of XX' after logical pooling                   22.90096 

Large value of kappa (>1000) indicate that rows are approximately proportional which is not good

  Conditional   Log-Likelihood: 47237.51    ;  np: 44 ;  AICc: -94387.01 

  Code/Message from optimization is:  1 singular convergence (7) 

Estimates
             pool1  pool2  pool3 pool4 pool5 pool6 psi cap.prob exp factor
pool.123       4.8    2.5    0.8   1.1   0.0   0.0 130    0.038       25.4
pool.123      14.4    8.8   10.9   1.1   3.9   0.0 330    0.038       25.4
pool.123       8.4   13.9   31.9   9.8   1.3   1.4 790    0.038       25.4
pool.456       1.2   17.1   30.1  13.2  12.1   1.5 667    0.033       29.2
pool.456       0.0    1.3    6.0   8.8   4.0   0.0 309    0.033       29.2
pool.456       0.0    0.0    0.0   0.0   0.0   1.5  65    0.033       29.2
est unmarked 739.0 1177.0 2160.0 948.0 603.0 126.0   0       NA         NA
             Pop Est
pool.123        3642
pool.123        9685
pool.123       22695
pool.456       22445
pool.456        9939
pool.456        1994
est unmarked   70399

SE of above estimates
             pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
pool.123       2.3   1.7   0.8   1.1   0.0   0.0 11.4    0.007        4.8
pool.123       3.8   3.1   2.9   1.1   2.2   0.0 18.2    0.007        4.8
pool.123       3.0   3.7   4.7   3.0   1.3   1.3 28.1    0.007        4.8
pool.456       1.3   3.9   4.4   3.2   2.9   1.2 25.8    0.006        5.8
pool.456       0.0   1.3   2.1   2.8   2.1   0.0 17.6    0.006        5.8
pool.456       0.0   0.0   0.0   0.0   0.0   1.2  8.1    0.006        5.8
est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
             Pop Est
pool.123         664
pool.123        1767
pool.123        4140
pool.456        4299
pool.456        1904
pool.456         382
est unmarked    4552


Chisquare gof cutoff  : 0.1 
Chisquare gof value   : 12.99086 
Chisquare gof df      : 4 
Chisquare gof p       : 0.01132054 

Here the stratified-Petersen estimate of the total number of smolts passing the first sampling station is 70,399 with a standard error of 70,399. which is a slight reduction from the unpooled estimates.

The estimates seem more sensible.

8.2.7.3 Pooling to a single row and complete pooling

You can pool to a single row (and multiple columns) or a single row and single column both of which are equivalent to the pooled Petersen estimator. The code and output follow:

Using nlminb to find conditional MLE
outer mgc:  5554.604 
outer mgc:  3255.445 
outer mgc:  686.8353 
outer mgc:  85.66413 
outer mgc:  12.0596 
outer mgc:  3.803003 
outer mgc:  0.6639956 
outer mgc:  0.2609515 
outer mgc:  0.09867538 
outer mgc:  0.03679076 
outer mgc:  0.01360926 
outer mgc:  0.005017272 
outer mgc:  0.001859472 
outer mgc:  0.0002086563 
outer mgc:  1.313641e-05 
Convergence codes from nlminb  1 singular convergence (7) 
Finding conditional estimate of N

Model Name: A single row 
   Date of Fit: 2025-10-24 15:54 
   Version of OPEN SPAS used : SPAS-R 2025.2.1 
 
Raw data 
     a    b    c   d   e   f    
01   4    2    1   1   0   0 130
02  12    7   14   1   3   0 330
03   7   11   41   9   1   1 790
04   1   13   40  12   9   1 667
05   0    1    8   8   3   0 309
06   0    0    0   0   0   1  65
   744 1187 2136 951 608 127   0

Row pooling setup : 1 1 1 1 1 1 
Col pooling setup : 1 2 3 4 5 6 
Physical pooling  : FALSE 
Theta pooling     : FALSE 
CJS pooling       : FALSE 


Chapman estimator of population size  70135  (SE  4503  )
 

Raw data AFTER PHYSICAL (but not logical) POOLING 
       pool1 pool2 pool3 pool4 pool5 pool6    
pool.1     4     2     1     1     0     0 130
pool.1    12     7    14     1     3     0 330
pool.1     7    11    41     9     1     1 790
pool.1     1    13    40    12     9     1 667
pool.1     0     1     8     8     3     0 309
pool.1     0     0     0     0     0     1  65
         744  1187  2136   951   608   127   0

Condition number of XX' where X= (physically) pooled matrix is  4121.288 
Condition number of XX' after logical pooling                   1 

Large value of kappa (>1000) indicate that rows are approximately proportional which is not good

  Conditional   Log-Likelihood: 47237.43    ;  np: 43 ;  AICc: -94388.87 

  Code/Message from optimization is:  1 singular convergence (7) 

Estimates
             pool1  pool2  pool3 pool4 pool5 pool6 psi cap.prob exp factor
pool.1         4.5    2.6    0.8   1.1   0.0   0.0 130    0.036       27.1
pool.1        13.6    8.9   10.7   1.1   4.2   0.0 330    0.036       27.1
pool.1         8.0   14.0   31.4  10.1   1.4   1.5 790    0.036       27.1
pool.1         1.1   16.6   30.6  13.5  12.5   1.5 667    0.036       27.1
pool.1         0.0    1.3    6.1   9.0   4.2   0.0 309    0.036       27.1
pool.1         0.0    0.0    0.0   0.0   0.0   1.5  65    0.036       27.1
est unmarked 741.0 1178.0 2160.0 947.0 602.0 125.0   0       NA         NA
             Pop Est
pool.1          3883
pool.1         10326
pool.1         24198
pool.1         20906
pool.1          9257
pool.1          1857
est unmarked   70426

SE of above estimates
             pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
pool.1         2.1   1.8   0.8   1.1   0.0   0.0 11.4    0.002        1.9
pool.1         3.0   3.1   2.8   1.1   2.2   0.0 18.2    0.002        1.9
pool.1         2.6   3.6   4.4   2.9   1.3   1.3 28.1    0.002        1.9
pool.1         1.1   3.8   4.4   3.2   2.9   1.3 25.8    0.002        1.9
pool.1         0.0   1.3   2.1   2.8   2.2   0.0 17.6    0.002        1.9
pool.1         0.0   0.0   0.0   0.0   0.0   1.3  8.1    0.002        1.9
est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
             Pop Est
pool.1           262
pool.1           696
pool.1          1632
pool.1          1410
pool.1           624
pool.1           125
est unmarked    4545


Chisquare gof cutoff  : 0.1 
Chisquare gof value   : 13.09369 
Chisquare gof df      : 5 
Chisquare gof p       : 0.0225164 

Now the estimated abundance of 70,426 with a standard error of 4,545. which is a slight reduction from the unpooled estimates.

8.2.7.4 Comparing the different poolings:

We can compare the fit using AICc in the usual way (Table 23)

spas.aictab <- Petersen::LP_AICc(mod..1, mod..2, mod..3, mod..4)

Table 23: Comparision of model different SPAS poolings

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
Pooling every second row	47,242.37	46	8,256	-94,392.22	0.00	0.63
No restrictions	47,243.45	48	8,256	-94,390.32	1.89	0.24
A single row	47,237.43	43	8,256	-94,388.41	3.81	0.09
Pooling early vs. late	47,237.51	44	8,256	-94,386.53	5.69	0.04

Notice that even with a high degree of logical pooling, SPAS still estimates a movement probability for the entire recapture matrix. All that logical pooling does is enforce equality of the initial capture probabilities. It is possible to do physical row pooling where the recapture matrix is also reduced, but then you cannot compare different physical poolings using AIC.

We see that there is a slight preference for pooling every two rows vs no pooling, but the condition factor of the model with no pooling makes it less than ideal.

We can also create a report of the estimates and model average them in the usual way (Table 24).

Error in LP_SPAS_est(x): LP_SPAS argument must be the results of a call to fitting SPAS objectl

Table 24: Model averaged estimates of abundance from different SPAS poolings

Modnames	AICcWt	Estimate	SE
Pooling every second row	0.63	74,701	10,284
No restrictions	0.24	78,159	14,676
A single row	0.09	70,426	4,545
Pooling early vs. late	0.04	70,399	4,552
Model averaged		74,986	11,252

Different column pooling will give the same fit and so cannot be compared. Refer to the vignette in the SPAS package on why column poolings cannot be compared for more details.

8.2.7.5 What does autopool suggest?

We can also try the autopooling options where SPAS pools rows and columns to ensure that sufficient data is present, but, (at this moment) does not check the singularity condition.

The autopooling function only suggests pooling the last 2 rows because the sample sizes are very small, but the condition factor is still large and so I would not consider this model.

8.2.8 Not all tagged fish have tags read in recovery strata

Schwarz, Andrews, and Link (1999) considered with model with the additional complexity that some of the fish recovered are seen to have tags, but the tags are not read. Consequently, it is not possible to distribute these fish back to the release stratum. In their example, fish pass through a fishway where it relatively easy to see if a fish has a tag, but only every 1/4 of fish are subsequently removed to read the tag and know the stratum of release.

Specialized software to fit the above model using the estimating equations has been written in R and is available upon request. This software also will fit simpler models where the tag-reading rate is assumed to be equal for all the recovery strata, where the tag application rate is equal for all release strata, or where the parameters can be modeled using covariates. In the middle model above, the estimate of the total population size is algebraically equal to the simple Petersen estimator, which is known to be consistent when all the initial capture probabilities are equal.

In the absence of specialized software for this experiment, approximate solutions can be found using software for the stratified Petersen with all tags read (SPAS) by initially distributing the unread tags into their respective rows in the same proportion as the \(m_{ij}\) to their column sums. Of course, the final variance estimates reported by SPAS will be too small.

8.2.9 Summary

The greatest problem that you will likely encounter while using SPAS is the near singularity of the recovery matrix. This is often unavoidable because there is no simple way to simplify the structure of the model and so all \(s \times t\) possible movement patterns must be accounted for. If the stratification is temporal, this imposes much additional structure that can be used (see next section). Consequently, I suspect you will find that more more than 3 or 4 rows or columns is about the limit that can successfully fit with SPAS in practice.

8.3 Temporal stratification

Temporal stratification is commonly used when estimating the run abundance of incoming adult salmon or outgoing juvenile salmon. The key difference from geographical stratification is that is is not possible for fish to be captured before they are releases leading to a large number of structural 0’s in the counts. The incoming/out coming abundance generally changes ‘slowly’ over time, so that the abundance in temporal stratum \(i\) has information on temporal stratum \(i+1\). Problems with the data are common, e.g. no fish released in some of the temporal strata, or no operations in other temporal strata, e.g., due to safety concerns leading to structural zeros.

Bonner and Schwarz (2011) developed a Bayesian model for temporally stratified sampling and created an \(R\) package, BTSPAS that analyses these studies. In this package, we provide a wrapper function to make it easier to use the BTSPAS package and to present results in a unified fashion. However, if finer control is wanted over the fitting procedures, the BTSPAS package can be used directly. Details about this model are presented in Bonner and Schwarz (2011) and in the vignettes that ship with the package. A synopsis of the theory is presented in Appendix D.

8.3.1 Sampling Protocol

Consider a study to estimate the number of outgoing smolts on a small river. The run of smolts extends over several weeks. As smolts migrate, they are captured and marked with individually numbered tags and released at the first capture location using, for example, a fishwheel. The migration continues, and a second fishwheel takes a second sample several kilometers down stream. At the second fishwheel, the captures consist of a mixture of marked (from the first fishwheel) and unmarked fish.

The efficiency of the fishwheels varies over time in response to stream flow, run size passing the wheel and other uncontrollable events. So it is unlikely that the capture probabilities are equal over time at either location, i.e. are heterogeneous over time.

We suppose that we can temporally stratify the data into, for example, weeks, where the capture-probabilities are (mostly) homogeneous at each wheel in each week.

8.3.2 Data structure

The same data structure as previously seen is also used, except the capture history is modified to account for potentially many temporal strata as follows:

• xx..yy represents a capture_history where xx and yy are the temporal stratum (e.g., julian week) and ‘..’ separates the two temporal strata. If a fish is released in temporal stratum and never captured again, then yy is set to 0; if a fish is newly captured in temporal stratum yy, then xx is set to zero.
• frequency variable for the number of fish with this capture history
• other covariates for this temporal stratum yy.

It is not possible to model the initial capture probability or to separate fish by stratification variables. These have rarely found to be useful in the temporally stratified models.

Some care is needed to properly account for 0 or missing values. For temporal strata with 0 releases, the BTSPAS wrappers will automatically impute 0 for the number of releases and recaptures if no capture history records are present. However, no imputation is done for strata is missing \(u_2\) values because this could represent a case were the trap was not running, rather than the trap did not capture any unmarked fish. The summary statistics should be be carefully checked when using this wrappers.

8.3.3 Key assumptions

The new key assumptions for this method are:

• sampling at second station is for the entire period of the stratum It is untrue, the user may need may need to impute and deal with the additional uncertainty
• all marked fish move at same rate as unmarked fish
• there is no fall back of tagged fish, i.e., after marking the marked fish continue on their migration in same fashion as unmarked fish.
• catchability of marked fish represents the catchability of the unmarked fish

Unfortunately, for most of these assumptions, there is little information in the data that help detect violations. Gross violations of this model can be detected from the goodness-of-fit plots presented in Bonner and Schwarz (2011).

There are two common cases

• Fish are stratified into temporal strata, and fish released in temporal stratum \(i\), move together and are captured together in a single future temporal stratum. This is called the Diagonal Model.
• Fish are stratified into temporal strata, but fish released in temporal stratum \(i\) can be captured in a number of future temporal strata. This is the called the NonDiagonal Case

The BTSPAS package also has other functions for dealing with multiple age classes in the study, but these are not discussed in this document.

8.3.4 Diagonal Model

In the diagonal model, fish in.a marked-cohort tend to move together and so tend to be recaptured close in time as well. Suppose that fish captured and marked in each week tend to migrate together so that they are captured in a single subsequent stratum. For example, suppose that in each julian week \(j\), \(n_{1j}\) fish are marked and released above the rotary screw trap. Of these, \(m_{2j}\) are recaptured. All recaptures take place in the week of release, i.e. the matrix of releases and recoveries is diagonal. The \(n_{1j}\) and \(m_{2j}\) establish the capture efficiency of the second trap in julian week \(j\).

At the same time, \(u_{2j}\) unmarked fish are captured at the screw trap.

We label the releases as \(n_{1i}\) to indicate this is the “first” capture and release of a cohort of fish, and \(m_{2i}\) or \(u_{2i}\) to indicate that this is “second” event where tagged fish are recaptured, and unmarked fish are newly capture.

Here is an example of data collected under this protocol:

data(data_btspas_diag1)
head(data_btspas_diag1)

  cap_hist  freq  logflow
1   00..04 14587 6.564691
2   04..00  1414 6.564691
3   04..04    51 6.564691
4   00..05  2854 7.077220
5   05..00  1189 7.077220
6   05..05   146 7.077220

In temporal stratum 4, 1465 fish were released at the first fishwheel, of which 51 were recaptured in the same week at the second fishwheel, and 1414 were never seen again. An additional 14587 unmarked fish were captured at the second fishwheel.

This can be summarized in a matrix format (up to week 10):

    fw2
fw1      0     4     5     6     7     8     9    10    11    12
  0      0 14587  2854  1027  1945  2855  1323   933 57549 14846
  4   1414    51     0     0     0     0     0     0     0     0
  5   1189     0   146     0     0     0     0     0     0     0
  6    180     0     0    17     0     0     0     0     0     0
  7   1167     0     0     0   168     0     0     0     0     0
  8   1581     0     0     0     0    72     0     0     0     0
  9    880     0     0     0     0     0    43     0     0     0
  10   582     0     0     0     0     0     0    28     0     0
  11  7672     0     0     0     0     0     0     0   297     0
  12  5695     0     0     0     0     0     0     0     0   313

The first row (corresponding to fw1=0) are the number of unmarked fish recovered at the second fish wheel in the temporal strata. The first column (corresponding to fw2=0) are the number of released marked fish that were never seen again.

We can also get the summary statistics (\(n_1\), \(m_2\), and \(u_2\)) for this data:

# compute n, m,u
nmu <- Petersen::cap_hist_to_n_m_u(data_btspas_diag1)
temp <- data.frame(time=nmu$..ts, n1=nmu$n1, m2=nmu$m2, u2=nmu$u2)
temp

   time   n1  m2    u2
1     4 1465  51 14587
2     5 1335 146  2854
3     6  197  17  1027
4     7 1335 168  1945
5     8 1653  72  2855
6     9  923  43  1323
7    10  610  28   933
8    11 7969 297 57549
9    12 6008 313 14846
10   13 3770 151  5291
11   14 4854 309  9249
12   15 3350 376  4615
13   16 1062 110  1433
14   17  346  40   446
15   18   88  12   120
16   19   20   1    23

There are no missing values

The theory in Appendix D, assumes that a single spline curve can be fit across all strata, but problems can arise in fitting this model to the data. In some cases, there are obvious breaks in the pattern of abundance over time that need to be accounted for in the model. F Attempting to fit a smooth curve across all strata ignores these jumps and so it is necessary to allow for breaks in the fitted spline.

It may occur that no marked fish (i.e., \(n_{1i}\) =0 for some \(i\)) are released in a particular stratum because no fish were available or because of logistical constraints. This would imply that a simple Petersen estimator for this stratum cannot be computed because no estimate of the capture probability is available. However, the Bayesian method will impute a range of capture-probabilities based on the capture probabilities in other strata, the shape of the spline curve, and the observed number of unmarked fish captured. The final estimate of abundance will (automatically) incorporate the uncertainty for this imputed capture probability.

If no data could be collected for a particular stratum (i.e., all of \(n_{1i}=0\), \(m_{2i}=0\), and \(u_{2i}=0\)), the spline will “impute” a value for the run size and capture probability in that stratum given the shape of the spline and the variability in individual run sizes about the spline; and will “impute” a value for the capture probability given the range of capture probabilities in the other strata. The final estimate of abundance will (automatically) incorporate the uncertainty for this imputed values.

While it is possible to interpolate for several strata in a row, there is of course, no information on the shape of the underlying spline during these missed strata, and so the results should be interpreted with care.

The Bayesian model assumes that sampling occurs throughout the temporal stratum. However, in some strata, sampling may take place in part of the stratum (i.e., in 3 of 7 days in a week). This causes no theoretical problem for estimation of the capture probability as the capture probability is assumed to be equal over the entire week so estimates of the probability of capture at the second trap based on 3 days may have poor precision but remain unbiased. However, the number of unmarked fish needs to be adjusted for the partial sampling during the stratum. We assume that the user has made this adjustment, and no accounting of the uncertainty in this adjustment will be made.

In some cases, the recapture effort varies over the course of a temporal stratum. For example, two rotary screw traps may be operating on Monday, and then only one trap is operating on Tuesday, etc. Unfortunately, this type of problem CANNOT be adequately dealt with by the spline (or any other method) that uses batch marks. The problem is that differing effort during a stratum (week) results in heterogeneity of catchability during the week, e.g., the catchability on days when two traps are operating is likely larger than the catchability on days when only one trap is operating. If a batch mark is used, the data is pooled over the stratum (week) and it is impossible to separate out catches according to how many traps are operating.

As for the pooled-Petersen estimator, this will likely result in estimates with low bias, but the precision of the estimates for these strata will be mis-reported, i.e., the standard deviations of the estimated run sizes for these strata will be understated. While there is no way to assess the extent of the problem (other than via simulations), it is hoped that the stratification into weekly strata will resolve most of the underreporting of the precision by the pooled-Petersen estimator and that any remaining understatement is not material.

The two key advantages of the Bayesian approach are:

• accounting for missing data when sampling could not take place
• the self-adjusting performance of the method. If sample sizes are large in each temporal stratum, then very little sharing of information is needed across the strata and very little smoothing of the run distribution is done. However, in case with small sample sizes, more sharing of information across strata occurs and more smoothing is done on the run shape.

Full details are presented in Schwarz and Bonner (2011) and the vignettes of the BTSPAS package.

8.3.4.1 Preliminary screening of the data

A pooled-Petersen estimator would add all of the marked, recaptured and unmarked fish to give an estimate of 1,987,456 (SE 41,322) fish but can the estimate be trusted?

Let us first examine a plot of the estimated capture efficiency at the second trap for each set of releases (Figure 18).

Figure 18: Empirical capture probabilities

There are several unusual features

• There appears to be heterogeneity in the capture probabilities across the season.
• In some temporal strata, the number of marked fish released and recaptured is very small which lead to estimates with poor precision.

Similarly, let us look at the pattern of unmarked fish captured at the second trap (Figure 19).

Figure 19: Observed number of unmarked recaptures

The number of unmarked fish captured suddenly jumps by several orders of magnitude (remember the above plot is on the log() scale) in temporal stratum 11. This jump corresponds to releases of hatchery fish into the system.

Finally, let us look at the individual estimates for each stratum found by computing a Petersen estimator for abundance of unmarked fish for each individual stratum (Figure 20).

Figure 20: Estimated log(total unmarked) by julian week

We see:

• The sudden jumps in abundance due to the hatchery releases is apparent
• There is a fairly regular pattern in abundance with a slow increase until the first hatchery release, followed by a steady decline.

8.3.4.2 Fitting the basic BTSPAS diagonal model

The BTSPAS package attempts to strike a balance between the completely pooled Petersen estimator and the completely stratified Petersen estimator. In the former, capture probabilities are assumed to equal for all fish in all strata, while in the latter, capture probabilities are allowed to vary among strata in no structured way. Furthermore, fish populations often have a general structure to the run, rather than arbitrarily jumping around from stratum to stratum.

The BTSPAS package also has additional features and options:

• the user to use covariates to explain some of the variation in the capture probabilities.
• if \(u_2\) is missing for any stratum, the program will use the spline to interpolate the number of unmarked fish in the population for the missing stratum.
• if \(n_1\) and \(m_2\) are 0, then these strata provide no information towards recapture probabilities. This is useful when no release take place in a stratum (e.g. trap did not run) and so you need ‘dummy’ values as placeholders. Of course if \(n_1>0\) and \(m_2=0\), this provides information that the capture probability may be small. If \(n_1=0\) and \(m_2>0\), this is an error (recoveries from no releases).
• the program allows you specify break points in the underlying spline to account for external events. We was in the above example, that hatchery fish were released at in julian weeks 23 and 40 resulting in sudden jump in abundance. The \(\textit{jump.after}\) parameter gives the julian weeks just BEFORE the sudden jump, i.e. the spline is allowed to jump AFTER the julian weeks in jump.after.

If unfortunate events happen where, for example, no fish could be released, or the second fish wheel is not running in a week, the data should be carefully modified to ensure that missing values are not replaced by 0’s.

The Petersen package function BTSPAS_Diag_fit() is a wrapper to the corresponding function in the BTSPAS package that takes the (modified for bad events) data file, a model for the mean capture probabilities at the second temporal stratum (default is a common mean), and identification of break points in the underlying spline, and then call the function in the BTSPAS package, and formats the returned data structure to match the returning structure for other functions in this package.

If finer control over the fitting process is needed, the BTSPAS package should be called directly – please consult the vignettes that come with the BTSPAS package.

We fit the model”

BTSPAS.diag.fit1 <- Petersen::LP_BTSPAS_fit_Diag(
         data_btspas_diag1,
         p_model=~1,
         jump.after=10,
         InitialSeed=23943242
  )

and look at the summary information on the fit:

The output object contains all of the results and can be saved for later interrogations. This is useful if the run takes considerable time (e.g. overnight) and you want to save the results for later processing.

As noted previously, model comparisons are not easily done using Bayesian methods except for perhaps using DIC (but this is often computed at the incorrect focus of the model). Furthermore, the BTSPAS model are self adjusting and usually the default fit is sufficient.

The final BTSPAS object returned by the fit has many components and contain summary tables, plots, and the the like:

names(BTSPAS.diag.fit1)

[1] "summary"     "data"        "p_model"     "p_model_cov" "jump.after" 
[6] "InitialSeed" "name_model"  "fit"         "datetime"   

names(BTSPAS.diag.fit1$fit)

 [1] "n.chains"           "n.iter"             "n.burnin"          
 [4] "n.thin"             "n.keep"             "n.sims"            
 [7] "sims.array"         "sims.list"          "sims.matrix"       
[10] "summary"            "mean"               "sd"                
[13] "median"             "root.short"         "long.short"        
[16] "dimension.short"    "indexes.short"      "last.values"       
[19] "program"            "model.file"         "isDIC"             
[22] "DICbyR"             "pD"                 "DIC"               
[25] "model"              "parameters.to.save" "plots"             
[28] "runTime"            "report"             "data"              

The plots sub-object contains many plots:

[1] "init.plot"         "fit.plot"          "logitP.plot"      
[4] "acf.Utot.plot"     "post.UNtot.plot"   "gof.plot"         
[7] "trace.logitP.plot" "trace.logU.plot"  

In particular, it contains plots of the initial spline fit (init.plot), the final fitted spline (fit.plot), the estimated capture probabilities (on the logit scale) (logitP.plot), plots of the distribution of the posterior sample for the total unmarked and marked fish (post.UNtot.plot) and model diagnostic plots (goodness of fit (gof.plot), trace (trace…plot), and autocorrelation plots (act.Utot.plot).

These plots are all created using the ggplot2 packages, so the user can modify the plot (e.g. change titles etc).

The BTSPAS program also creates a report, which includes information about the data used in the fitting, the pooled- and stratified-Petersen estimates, a test for pooling, and summaries of the posterior. Only the first few lines are shown below:

 [1] "Time Stratified Petersen with Diagonal recaptures and error in smoothed U -  Mon May 27 20:23:32 2024"
 [2] "Version: 2024-05-09 "                                                                                 
 [3] ""                                                                                                     
 [4] ""                                                                                                     
 [5] ""                                                                                                     
 [6] "  Results "                                                                                           
 [7] ""                                                                                                     
 [8] "*** Raw data *** "                                                                                    
 [9] "      time   n1  m2    u2 logitPcov[1]"                                                               
[10] " [1,]    4 1465  51 14587            1"                                                               
[11] " [2,]    5 1335 146  2854            1"                                                               
[12] " [3,]    6  197  17  1027            1"                                                               
[13] " [4,]    7 1335 168  1945            1"                                                               
[14] " [5,]    8 1653  72  2855            1"                                                               
[15] " [6,]    9  923  43  1323            1"                                                               
[16] " [7,]   10  610  28   933            1"                                                               
[17] " [8,]   11 7969 297 57549            1"                                                               
[18] " [9,]   12 6008 313 14846            1"                                                               
[19] "[10,]   13 3770 151  5291            1"                                                               
[20] "[11,]   14 4854 309  9249            1"                                                               

The fitted spline curve to the number of unmarked fish available in each recovery sample is shown in ?@fig-btspas-diag1-fit-plot

$data
   time     logUi      logU   logUlcl   logUucl    spline
1     4 12.927686 12.858400 12.621356 13.115329 11.546632
2     5 10.164578 10.199596 10.044304 10.357170 10.983029
3     6  9.338304  9.561254  9.152286 10.007612 10.627844
4     7  9.641816  9.673944  9.528708  9.824896 10.392922
5     8 11.078274 11.047969 10.833476 11.274499 10.197187
6     9 10.234017 10.220988  9.956258 10.510632  9.987867
7    10  9.888913  9.874247  9.555136 10.208115  9.719269
8    11 14.246881 14.243708 14.136136 14.358751 14.277170
9    12 12.557340 12.561275 12.454094 12.669492 13.056713
10   13 11.785432 11.783149 11.626268 11.939636 12.029582
11   14 11.883777 11.882899 11.775472 11.995667 11.125502
12   15 10.622351 10.629829 10.533924 10.730142 10.274199
13   16  9.528483  9.553578  9.376165  9.735739  9.405964
14   17  8.241189  8.330916  8.039542  8.631933  8.453353
15   18  6.730651  7.050205  6.533267  7.608104  7.349487
16   19  5.575949  5.942875  5.128041  6.817782  6.027489

$layers
$layers[[1]]
mapping: y = ~logUi 
geom_point: na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 

$layers[[2]]
mapping: y = ~logU 
geom_point: na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 

$layers[[3]]
mapping: y = ~logU 
geom_line: na.rm = FALSE, orientation = NA
stat_identity: na.rm = FALSE
position_identity 

$layers[[4]]
mapping: ymin = ~logUlcl, ymax = ~logUucl 
geom_errorbar: na.rm = FALSE, orientation = NA, width = 0.1
stat_identity: na.rm = FALSE
position_identity 

$layers[[5]]
mapping: y = ~spline 
geom_line: na.rm = FALSE, orientation = NA
stat_identity: na.rm = FALSE
position_identity 


$scales
<ggproto object: Class ScalesList, gg>
    add: function
    add_defaults: function
    add_missing: function
    backtransform_df: function
    clone: function
    find: function
    get_scales: function
    has_scale: function
    input: function
    map_df: function
    n: function
    non_position_scales: function
    scales: list
    set_palettes: function
    train_df: function
    transform_df: function
    super:  <ggproto object: Class ScalesList, gg>

$guides
<Guides[0] ggproto object>

<empty>

$mapping
$x
<quosure>
expr: ^time
env:  0x7fc5b38a8298

attr(,"class")
[1] "uneval"

$theme
list()

$coordinates
<ggproto object: Class CoordCartesian, Coord, gg>
    aspect: function
    backtransform_range: function
    clip: on
    default: TRUE
    distance: function
    draw_panel: function
    expand: TRUE
    is_free: function
    is_linear: function
    labels: function
    limits: list
    modify_scales: function
    range: function
    render_axis_h: function
    render_axis_v: function
    render_bg: function
    render_fg: function
    reverse: none
    setup_data: function
    setup_layout: function
    setup_panel_guides: function
    setup_panel_params: function
    setup_params: function
    train_panel_guides: function
    transform: function
    super:  <ggproto object: Class CoordCartesian, Coord, gg>

$facet
<ggproto object: Class FacetNull, Facet, gg>
    attach_axes: function
    attach_strips: function
    compute_layout: function
    draw_back: function
    draw_front: function
    draw_labels: function
    draw_panel_content: function
    draw_panels: function
    finish_data: function
    format_strip_labels: function
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    init_scales: function
    map_data: function
    params: list
    set_panel_size: function
    setup_data: function
    setup_panel_params: function
    setup_params: function
    shrink: TRUE
    train_scales: function
    vars: function
    super:  <ggproto object: Class FacetNull, Facet, gg>

$plot_env
<environment: 0x7fc5b38a8298>

$layout
<ggproto object: Class Layout, gg>
    coord: NULL
    coord_params: list
    facet: NULL
    facet_params: list
    finish_data: function
    get_scales: function
    layout: NULL
    map_position: function
    panel_params: NULL
    panel_scales_x: NULL
    panel_scales_y: NULL
    render: function
    render_labels: function
    reset_scales: function
    resolve_label: function
    setup: function
    setup_panel_guides: function
    setup_panel_params: function
    train_position: function
    super:  <ggproto object: Class Layout, gg>

$labels
$labels$y
[1] "log(U[i]) + 95% credible interval"

$labels$x
[1] "Time Index\nOpen/closed circles - initial and final estimates"

$labels$title
[1] ""

$labels$subtitle
[1] "Fitted spline curve with 95% credible intervals for estimated log(U[i])"

$labels$ymin
[1] "logUlcl"

$labels$ymax
[1] "logUucl"


attr(,"class")
[1] "gg"     "ggplot"

The jump in the spline when hatchery fish are released is evident. The actual number of unmarked fish is allowed to vary around the spline as shown below.

The distribution of the posterior sample for the total number unmarked and total abundance is available as shown in ?@fig-btspas-diag1-postUN

$data
      parm  sample
1     Utot 2783519
2     Utot 2768535
3     Utot 2699240
4     Utot 2550164
5     Utot 2673304
6     Utot 2836941
7     Utot 2781521
8     Utot 2727428
9     Utot 2479725
10    Utot 2493288
11    Utot 2761439
12    Utot 2800276
13    Utot 2588862
14    Utot 2907390
15    Utot 2860403
16    Utot 2808518
17    Utot 2727836
18    Utot 2834408
19    Utot 2891563
20    Utot 2789477
21    Utot 2761644
22    Utot 2735427
23    Utot 2699897
24    Utot 2750135
25    Utot 2767245
26    Utot 2652881
27    Utot 2622503
28    Utot 2813029
29    Utot 2678718
30    Utot 2699907
31    Utot 2745041
32    Utot 2735805
33    Utot 2819504
34    Utot 2670227
35    Utot 2590954
36    Utot 2774458
37    Utot 2824170
38    Utot 2739734
39    Utot 2914236
40    Utot 2634506
41    Utot 2924845
42    Utot 2830612
43    Utot 2782706
44    Utot 2596148
45    Utot 2738677
46    Utot 2926443
47    Utot 2508027
48    Utot 2654390
49    Utot 2665657
50    Utot 2618884
51    Utot 2720681
52    Utot 2473617
53    Utot 2732428
54    Utot 2809322
55    Utot 2818914
56    Utot 2980773
57    Utot 2839010
58    Utot 2700822
59    Utot 2796278
60    Utot 2875657
61    Utot 2670858
62    Utot 2485682
63    Utot 2794367
64    Utot 2613473
65    Utot 2772038
66    Utot 2752361
67    Utot 2692578
68    Utot 2806932
69    Utot 2684958
70    Utot 2664413
71    Utot 2755502
72    Utot 2846104
73    Utot 2814106
74    Utot 2866886
75    Utot 2818553
76    Utot 2566972
77    Utot 2608723
78    Utot 2773979
79    Utot 2779293
80    Utot 2869542
81    Utot 2722297
82    Utot 2808504
83    Utot 2645385
84    Utot 2746960
85    Utot 2826707
86    Utot 2589867
87    Utot 2785830
88    Utot 2793419
89    Utot 2615393
90    Utot 2705284
91    Utot 2710700
92    Utot 2840347
93    Utot 2635031
94    Utot 2841174
95    Utot 2829757
96    Utot 2689310
97    Utot 2841475
98    Utot 2563537
99    Utot 2702711
100   Utot 2744809
101   Utot 2752633
102   Utot 2901935
103   Utot 2620158
104   Utot 2743359
105   Utot 2650614
106   Utot 2715164
107   Utot 2871801
108   Utot 2645348
109   Utot 2836835
110   Utot 2690552
111   Utot 2774549
112   Utot 2800272
113   Utot 2600724
114   Utot 2685610
115   Utot 2760589
116   Utot 2645236
117   Utot 2469391
118   Utot 2736737
119   Utot 2631140
120   Utot 2819857
121   Utot 2703776
122   Utot 2614727
123   Utot 2724698
124   Utot 2830653
125   Utot 2635688
126   Utot 2829779
127   Utot 2810537
128   Utot 2735091
129   Utot 2654128
130   Utot 2590992
131   Utot 2678034
132   Utot 2563317
133   Utot 2909645
134   Utot 2774493
135   Utot 2887912
136   Utot 2655483
137   Utot 2760335
138   Utot 2692196
139   Utot 2748306
140   Utot 2652051
141   Utot 2740765
142   Utot 2689817
143   Utot 2633003
144   Utot 2749812
145   Utot 2772198
146   Utot 2674679
147   Utot 2637327
148   Utot 2649267
149   Utot 2847512
150   Utot 2695332
151   Utot 2788560
152   Utot 2568017
153   Utot 2838069
154   Utot 2633543
155   Utot 2720079
156   Utot 2818650
157   Utot 2557862
158   Utot 2659965
159   Utot 2713494
160   Utot 2639655
161   Utot 2633698
162   Utot 2625115
163   Utot 2648194
164   Utot 2791566
165   Utot 2478984
166   Utot 2814107
167   Utot 2603648
168   Utot 2736707
169   Utot 2665362
170   Utot 2651430
171   Utot 2802891
172   Utot 2749168
173   Utot 2725972
174   Utot 2900267
175   Utot 2579692
176   Utot 2822909
177   Utot 2629800
178   Utot 2730496
179   Utot 2590638
180   Utot 2614621
181   Utot 2585105
182   Utot 2708589
183   Utot 2873435
184   Utot 2774325
185   Utot 2546845
186   Utot 2884504
187   Utot 2695765
188   Utot 2648814
189   Utot 2855022
190   Utot 2691155
191   Utot 2680015
192   Utot 2591987
193   Utot 2606228
194   Utot 2844293
195   Utot 2709851
196   Utot 2836222
197   Utot 2591797
198   Utot 2673028
199   Utot 2828387
200   Utot 2746360
201   Utot 2754182
202   Utot 2612969
203   Utot 2639064
204   Utot 2728507
205   Utot 2807091
206   Utot 2715870
207   Utot 2719037
208   Utot 2544774
209   Utot 2626019
210   Utot 2440450
211   Utot 2605553
212   Utot 2678174
213   Utot 2781596
214   Utot 2738276
215   Utot 2530347
216   Utot 2590570
217   Utot 2667736
218   Utot 2656669
219   Utot 2673728
220   Utot 2727785
221   Utot 2793254
222   Utot 2627504
223   Utot 2575111
224   Utot 2746400
225   Utot 2845999
226   Utot 2635319
227   Utot 2682882
228   Utot 2768540
229   Utot 2728081
230   Utot 2862987
231   Utot 2832534
232   Utot 2788871
233   Utot 2572813
234   Utot 2633422
235   Utot 2857492
236   Utot 2797289
237   Utot 2819721
238   Utot 2724105
239   Utot 2623643
240   Utot 2740345
241   Utot 2714689
242   Utot 2875365
243   Utot 2719157
244   Utot 2759492
245   Utot 2835710
246   Utot 2838734
247   Utot 2670702
248   Utot 2774787
249   Utot 2801796
250   Utot 2503839
251   Utot 2712191
252   Utot 2713260
253   Utot 2715185
254   Utot 2648538
255   Utot 2897898
256   Utot 2671591
257   Utot 2719840
258   Utot 2681363
259   Utot 2712726
260   Utot 2564976
261   Utot 2777289
262   Utot 2640739
263   Utot 2806004
264   Utot 2692169
265   Utot 2766618
266   Utot 2692840
267   Utot 2725621
268   Utot 3053133
269   Utot 2674725
270   Utot 2756676
271   Utot 2568448
272   Utot 2622331
273   Utot 2683296
274   Utot 2643139
275   Utot 2717427
276   Utot 2696430
277   Utot 2728714
278   Utot 2659228
279   Utot 2859506
280   Utot 2571176
281   Utot 2898701
282   Utot 2630513
283   Utot 2694778
284   Utot 2834354
285   Utot 2853603
286   Utot 2578605
287   Utot 2733628
288   Utot 2753441
289   Utot 2781804
290   Utot 2781795
291   Utot 2829167
292   Utot 2942377
293   Utot 2820857
294   Utot 2540411
295   Utot 2744184
296   Utot 2727833
297   Utot 2607609
298   Utot 2707398
299   Utot 2797922
300   Utot 2775889
301   Utot 2521629
302   Utot 2816656
303   Utot 2805733
304   Utot 2778442
305   Utot 2805254
306   Utot 2880321
307   Utot 2772344
308   Utot 2798063
309   Utot 2506546
310   Utot 2943425
311   Utot 2809345
312   Utot 2542444
313   Utot 2865563
314   Utot 2736915
315   Utot 2675733
316   Utot 2705664
317   Utot 2759516
318   Utot 3058202
319   Utot 2741231
320   Utot 2642324
321   Utot 2911447
322   Utot 2735093
323   Utot 2746212
324   Utot 2999516
325   Utot 2706356
326   Utot 2844612
327   Utot 2575523
328   Utot 2622517
329   Utot 2659609
330   Utot 2920407
331   Utot 2908555
332   Utot 2791410
333   Utot 2841541
334   Utot 2703694
335   Utot 2657963
336   Utot 2744086
337   Utot 2677084
338   Utot 2795191
339   Utot 2701918
340   Utot 2951285
341   Utot 2509567
342   Utot 2602884
343   Utot 2722536
344   Utot 2541620
345   Utot 2792850
346   Utot 2681456
347   Utot 2889582
348   Utot 2956402
349   Utot 2759895
350   Utot 2991825
351   Utot 2732771
352   Utot 2861242
353   Utot 2758053
354   Utot 2959560
355   Utot 2734911
356   Utot 2725880
357   Utot 2740015
358   Utot 2719398
359   Utot 2955417
360   Utot 2737688
361   Utot 2674085
362   Utot 2739614
363   Utot 2708903
364   Utot 2711817
365   Utot 2765098
366   Utot 2712971
367   Utot 2717940
368   Utot 2761580
369   Utot 2551075
370   Utot 2895940
371   Utot 2978321
372   Utot 2838818
373   Utot 2744313
374   Utot 2612781
375   Utot 2837848
376   Utot 2562540
377   Utot 2534089
378   Utot 2748767
379   Utot 2794751
380   Utot 2651684
381   Utot 2811591
382   Utot 2655589
383   Utot 2888873
384   Utot 2734582
385   Utot 2844593
386   Utot 2763113
387   Utot 2726422
388   Utot 2752248
389   Utot 2924283
390   Utot 2707594
391   Utot 2664688
392   Utot 2642596
393   Utot 2831570
394   Utot 2788483
395   Utot 2610476
396   Utot 2546278
397   Utot 2738669
398   Utot 2861935
399   Utot 2712290
400   Utot 2637040
401   Utot 2721966
402   Utot 2609442
403   Utot 2722874
404   Utot 2818463
405   Utot 2746465
406   Utot 2740046
407   Utot 2546684
408   Utot 2627566
409   Utot 2756116
410   Utot 2568649
411   Utot 2770370
412   Utot 2607109
413   Utot 2659871
414   Utot 2796699
415   Utot 2639818
416   Utot 2927436
417   Utot 2603827
418   Utot 2769374
419   Utot 2802169
420   Utot 2716986
421   Utot 2699372
422   Utot 2743245
423   Utot 2562437
424   Utot 2652543
425   Utot 2738379
426   Utot 2658172
427   Utot 2712053
428   Utot 2618233
429   Utot 2630308
430   Utot 2702319
431   Utot 2725751
432   Utot 2769675
433   Utot 2689167
434   Utot 2719632
435   Utot 2743930
436   Utot 2740936
437   Utot 2613207
438   Utot 2712293
439   Utot 2563666
440   Utot 2773978
441   Utot 2813162
442   Utot 2737667
443   Utot 2746990
444   Utot 2505520
445   Utot 2762769
446   Utot 2740422
447   Utot 2754998
448   Utot 2687478
449   Utot 2898301
450   Utot 2617237
451   Utot 2655181
452   Utot 2756671
453   Utot 2845082
454   Utot 2638794
455   Utot 2869514
456   Utot 2674670
457   Utot 2703906
458   Utot 2692872
459   Utot 2658068
460   Utot 3031698
461   Utot 2792989
462   Utot 2587693
463   Utot 2835798
464   Utot 2664455
465   Utot 2707039
466   Utot 2746351
467   Utot 2746430
468   Utot 2774988
469   Utot 2473269
470   Utot 2938598
471   Utot 2587847
472   Utot 2793337
473   Utot 3001432
474   Utot 2577067
475   Utot 2807727
476   Utot 2688282
477   Utot 2615903
478   Utot 2858771
479   Utot 2588140
480   Utot 2639987
481   Utot 2731063
482   Utot 2611338
483   Utot 2641451
484   Utot 2663680
485   Utot 2897577
486   Utot 2809273
487   Utot 2605588
488   Utot 2619449
489   Utot 2663569
490   Utot 2651721
491   Utot 2848835
492   Utot 2710995
493   Utot 2781836
494   Utot 2684485
495   Utot 2716782
496   Utot 2924141
497   Utot 2775463
498   Utot 2843624
499   Utot 2784365
500   Utot 2685169
501   Utot 2761824
502   Utot 2538023
503   Utot 2725733
504   Utot 2707245
505   Utot 2739551
506   Utot 2988699
507   Utot 2834396
508   Utot 2716515
509   Utot 2804982
510   Utot 2681864
511   Utot 2723821
512   Utot 2725364
513   Utot 2747908
514   Utot 2689984
515   Utot 2882851
516   Utot 2555349
517   Utot 2508106
518   Utot 2806437
519   Utot 2725430
520   Utot 2715745
521   Utot 2760572
522   Utot 2592183
523   Utot 2789312
524   Utot 2654374
525   Utot 2744662
526   Utot 2688560
527   Utot 2726679
528   Utot 2668422
529   Utot 3114274
530   Utot 2821195
531   Utot 2879667
532   Utot 2761460
533   Utot 2756258
534   Utot 2694268
535   Utot 2622689
536   Utot 2663886
537   Utot 2762838
538   Utot 2694187
539   Utot 2771909
540   Utot 2794709
541   Utot 2790461
542   Utot 2687814
543   Utot 2790640
544   Utot 2588889
545   Utot 2910538
546   Utot 2608439
547   Utot 2811852
548   Utot 2731425
549   Utot 2669993
550   Utot 2580929
551   Utot 2721727
552   Utot 2719320
553   Utot 2686400
554   Utot 2805196
555   Utot 2619015
556   Utot 2666651
557   Utot 2649665
558   Utot 2625787
559   Utot 2696032
560   Utot 2648736
561   Utot 2659345
562   Utot 2667409
563   Utot 2911537
564   Utot 2796331
565   Utot 2577889
566   Utot 2743011
567   Utot 2664193
568   Utot 2706081
569   Utot 2711713
570   Utot 2668128
571   Utot 2649078
572   Utot 2768723
573   Utot 2789431
574   Utot 2752403
575   Utot 2718062
576   Utot 2739350
577   Utot 2693530
578   Utot 2728200
579   Utot 2653229
580   Utot 2693083
581   Utot 2775900
582   Utot 2685385
583   Utot 2757345
584   Utot 2526737
585   Utot 2872595
586   Utot 2651288
587   Utot 2747147
588   Utot 2595240
589   Utot 2681655
590   Utot 2791830
591   Utot 2655231
592   Utot 2739927
593   Utot 2639482
594   Utot 2557583
595   Utot 2819668
596   Utot 2736194
597   Utot 2742589
598   Utot 2767971
599   Utot 2734595
600   Utot 2675200
601   Utot 2726564
602   Utot 2994640
603   Utot 2768681
604   Utot 2675926
605   Utot 2709409
606   Utot 2716062
607   Utot 2748251
608   Utot 2890888
609   Utot 2695017
610   Utot 2766239
611   Utot 2686783
612   Utot 2642913
613   Utot 2700067
614   Utot 2593920
615   Utot 2697009
616   Utot 2752334
617   Utot 2855546
618   Utot 2866365
619   Utot 2707327
620   Utot 2692177
621   Utot 2606404
622   Utot 2840135
623   Utot 2838272
624   Utot 2531291
625   Utot 2572502
626   Utot 2699691
627   Utot 2663030
628   Utot 2702065
629   Utot 2773914
630   Utot 2490426
631   Utot 2725291
632   Utot 2805568
633   Utot 2743854
634   Utot 2689711
635   Utot 2655466
636   Utot 2797235
637   Utot 2725736
638   Utot 2691050
639   Utot 2640620
640   Utot 2857401
641   Utot 2762911
642   Utot 2773626
643   Utot 2963613
644   Utot 2684399
645   Utot 2647482
646   Utot 2601613
647   Utot 2731418
648   Utot 2694968
649   Utot 2900139
650   Utot 2616384
651   Utot 2575512
652   Utot 2660671
653   Utot 2745173
654   Utot 2812700
655   Utot 2735099
656   Utot 2440536
657   Utot 2835371
658   Utot 2623990
659   Utot 2729543
660   Utot 2818626
661   Utot 2754286
662   Utot 2702967
663   Utot 2720853
664   Utot 2792928
665   Utot 2757558
666   Utot 2756397
667   Utot 2675420
668   Utot 2579046
669   Utot 2732489
670   Utot 2703913
671   Utot 2733122
672   Utot 2761748
673   Utot 2838784
674   Utot 2716467
675   Utot 2827932
676   Utot 2580352
677   Utot 2648209
678   Utot 2720136
679   Utot 2797427
680   Utot 2592549
681   Utot 2732630
682   Utot 2669604
683   Utot 2914978
684   Utot 2748959
685   Utot 2576718
686   Utot 2921848
687   Utot 2874800
688   Utot 2726840
689   Utot 2750613
690   Utot 2836810
691   Utot 2574863
692   Utot 2751083
693   Utot 2822420
694   Utot 2729046
695   Utot 2746890
696   Utot 2627047
697   Utot 2726739
698   Utot 2737343
699   Utot 2650667
700   Utot 2628970
701   Utot 2659927
702   Utot 2636133
703   Utot 2603885
704   Utot 2734570
705   Utot 2799277
706   Utot 2660556
707   Utot 2687590
708   Utot 2835835
709   Utot 2742803
710   Utot 2799048
711   Utot 2739562
712   Utot 2621348
713   Utot 2630038
714   Utot 2770622
715   Utot 2781596
716   Utot 2604785
717   Utot 2818994
718   Utot 2685075
719   Utot 2775334
720   Utot 2613455
721   Utot 2735972
722   Utot 2447622
723   Utot 2657323
724   Utot 2765890
725   Utot 2647660
726   Utot 2671883
727   Utot 2682170
728   Utot 2603798
729   Utot 2594216
730   Utot 2722559
731   Utot 2813928
732   Utot 2764388
733   Utot 2530811
734   Utot 2572123
735   Utot 2608891
736   Utot 2539951
737   Utot 2737024
738   Utot 2786818
739   Utot 2656890
740   Utot 2605167
741   Utot 2632921
742   Utot 2761320
743   Utot 2526413
744   Utot 2533297
745   Utot 2627913
746   Utot 2601555
747   Utot 2553686
748   Utot 2814383
749   Utot 2816368
750   Utot 2663878
751   Utot 2699796
752   Utot 2674059
753   Utot 2698478
754   Utot 2797716
755   Utot 2722025
756   Utot 2688715
757   Utot 2783930
758   Utot 2635557
759   Utot 2806084
760   Utot 2717256
761   Utot 2691305
762   Utot 2640185
763   Utot 2873868
764   Utot 2619651
765   Utot 2642973
766   Utot 2604760
767   Utot 2804125
768   Utot 2732449
769   Utot 2791940
770   Utot 2861565
771   Utot 2620666
772   Utot 2680816
773   Utot 2803238
774   Utot 2980770
775   Utot 2645752
776   Utot 2561312
777   Utot 2663653
778   Utot 2559367
779   Utot 2605444
780   Utot 2559028
781   Utot 2622736
782   Utot 2703081
783   Utot 2672376
784   Utot 2721772
785   Utot 2704394
786   Utot 2614986
787   Utot 2823239
788   Utot 2653151
789   Utot 2687295
790   Utot 2792738
791   Utot 2658551
792   Utot 2764487
793   Utot 2753474
794   Utot 2716073
795   Utot 2676270
796   Utot 2606458
797   Utot 2685341
798   Utot 2761154
799   Utot 2753239
800   Utot 2703805
801   Utot 2777319
802   Utot 2750276
803   Utot 2673896
804   Utot 2808442
805   Utot 2779243
806   Utot 2637753
807   Utot 2846929
808   Utot 2719879
809   Utot 2703813
810   Utot 2804751
811   Utot 2615781
812   Utot 2557913
813   Utot 2817118
814   Utot 2625233
815   Utot 2636704
816   Utot 2626762
817   Utot 2727891
818   Utot 2701889
819   Utot 2622446
820   Utot 2687374
821   Utot 2871030
822   Utot 2604633
823   Utot 2535134
824   Utot 2805896
825   Utot 2803870
826   Utot 2672222
827   Utot 2876827
828   Utot 2739142
829   Utot 2673052
830   Utot 3029574
831   Utot 2838479
832   Utot 2702148
833   Utot 2640214
834   Utot 2814979
835   Utot 2709545
836   Utot 2763024
837   Utot 2556125
838   Utot 2692572
839   Utot 2592922
840   Utot 2519221
841   Utot 2782056
842   Utot 2614749
843   Utot 2684802
844   Utot 2661440
845   Utot 2689565
846   Utot 2771362
847   Utot 2725183
848   Utot 2672290
849   Utot 2681412
850   Utot 2696334
851   Utot 2688234
852   Utot 3035448
853   Utot 2605184
854   Utot 2610177
855   Utot 2693749
856   Utot 2632194
857   Utot 2741388
858   Utot 2670410
859   Utot 2908015
860   Utot 2688281
861   Utot 2762206
862   Utot 2733844
863   Utot 2597293
864   Utot 2710622
865   Utot 2769292
866   Utot 2725935
867   Utot 2669032
868   Utot 2710179
869   Utot 2772491
870   Utot 2833377
871   Utot 2660944
872   Utot 2758309
873   Utot 2719203
874   Utot 2692256
875   Utot 2627238
876   Utot 2689473
877   Utot 2592533
878   Utot 2831656
879   Utot 2757394
880   Utot 2658446
881   Utot 2665646
882   Utot 2695674
883   Utot 2623614
884   Utot 2893201
885   Utot 2607933
886   Utot 2721627
887   Utot 2713850
888   Utot 2631948
889   Utot 2836774
890   Utot 2771797
891   Utot 2533069
892   Utot 2610443
893   Utot 2634268
894   Utot 2663284
895   Utot 2733416
896   Utot 2791393
897   Utot 2599573
898   Utot 2572484
899   Utot 2759789
900   Utot 2823981
901   Utot 2750436
902   Utot 2692924
903   Utot 2697317
904   Utot 2774956
905   Utot 2558331
906   Utot 2671890
907   Utot 2794283
908   Utot 2611580
909   Utot 2723156
910   Utot 2796356
911   Utot 2642858
912   Utot 2679676
913   Utot 2823531
914   Utot 2915499
915   Utot 2586140
916   Utot 2712693
917   Utot 2703569
918   Utot 2813123
919   Utot 2606795
920   Utot 2638689
921   Utot 2847361
922   Utot 2690766
923   Utot 2652620
924   Utot 2743543
925   Utot 2635313
926   Utot 2655544
927   Utot 2804161
928   Utot 2716010
929   Utot 2841021
930   Utot 2784284
931   Utot 2723938
932   Utot 2631852
933   Utot 2759171
934   Utot 2625458
935   Utot 2611780
936   Utot 2834561
937   Utot 2640426
938   Utot 2661580
939   Utot 2707483
940   Utot 2667460
941   Utot 2589555
942   Utot 2790004
943   Utot 2675465
944   Utot 2884643
945   Utot 2771087
946   Utot 2563073
947   Utot 2605676
948   Utot 2619023
949   Utot 2665936
950   Utot 2583872
951   Utot 2645266
952   Utot 2593167
953   Utot 2660101
954   Utot 2626298
955   Utot 2632375
956   Utot 2634633
957   Utot 2790150
958   Utot 2713628
959   Utot 2679419
960   Utot 2844403
961   Utot 2773514
962   Utot 2685304
963   Utot 2726507
964   Utot 2691482
965   Utot 2880687
966   Utot 2697487
967   Utot 2732822
968   Utot 2777626
969   Utot 2841383
970   Utot 2715261
971   Utot 2811906
972   Utot 2694932
973   Utot 2727058
974   Utot 2744995
975   Utot 2582889
976   Utot 2713639
977   Utot 2671855
978   Utot 2672375
979   Utot 2600305
980   Utot 2719403
981   Utot 2734284
982   Utot 2630324
983   Utot 2834995
984   Utot 2735414
985   Utot 2755986
986   Utot 2622765
987   Utot 2837281
988   Utot 2573358
989   Utot 2684667
990   Utot 2613548
991   Utot 2920757
992   Utot 2881721
993   Utot 2690865
994   Utot 2610199
995   Utot 2856814
996   Utot 2719365
997   Utot 2597461
998   Utot 2553726
999   Utot 2783119
1000  Utot 2621186
1001  Utot 2782707
1002  Utot 2906067
1003  Utot 2765012
1004  Utot 2717018
1005  Utot 2690120
1006  Utot 2774637
1007  Utot 2713500
1008  Utot 2705912
1009  Utot 2566167
1010  Utot 2867455
1011  Utot 2614933
1012  Utot 2762378
1013  Utot 2717419
1014  Utot 2787793
1015  Utot 2615637
1016  Utot 2675711
1017  Utot 2748626
1018  Utot 2737252
1019  Utot 2681744
1020  Utot 2694222
1021  Utot 2803846
1022  Utot 2594828
1023  Utot 2628367
1024  Utot 2710938
1025  Utot 2762817
1026  Utot 2781078
1027  Utot 2879257
1028  Utot 2830944
1029  Utot 2736100
1030  Utot 2699324
1031  Utot 2640014
1032  Utot 2712389
1033  Utot 2678587
1034  Utot 2640681
1035  Utot 2999920
1036  Utot 2642502
1037  Utot 2590010
1038  Utot 2513155
1039  Utot 2725713
1040  Utot 2904395
1041  Utot 2768775
1042  Utot 2889818
1043  Utot 2599421
1044  Utot 2778339
1045  Utot 2757319
1046  Utot 2859236
1047  Utot 2630425
1048  Utot 2671414
1049  Utot 2585711
1050  Utot 2774734
1051  Utot 2752727
1052  Utot 2667403
1053  Utot 2769940
1054  Utot 2646004
1055  Utot 2841193
1056  Utot 2713482
1057  Utot 2852160
1058  Utot 2917426
1059  Utot 2642616
1060  Utot 2790668
1061  Utot 2784615
1062  Utot 2781066
1063  Utot 2707892
1064  Utot 2536376
1065  Utot 2739062
1066  Utot 2676557
1067  Utot 2770945
1068  Utot 2763876
1069  Utot 2637530
1070  Utot 2640259
1071  Utot 2772562
1072  Utot 2581730
1073  Utot 2543630
1074  Utot 2578880
1075  Utot 2671259
1076  Utot 2795322
1077  Utot 2712888
1078  Utot 2837163
1079  Utot 2704466
1080  Utot 2586090
1081  Utot 2719744
1082  Utot 2688925
1083  Utot 2640915
1084  Utot 2789929
1085  Utot 2909709
1086  Utot 2774542
1087  Utot 2721026
1088  Utot 2596814
1089  Utot 2684190
1090  Utot 2854748
1091  Utot 2542773
1092  Utot 2712222
1093  Utot 2608983
1094  Utot 2679958
1095  Utot 2580126
1096  Utot 2571913
1097  Utot 2735341
1098  Utot 2561599
1099  Utot 2868800
1100  Utot 2649379
1101  Utot 2870723
1102  Utot 2612035
1103  Utot 2654019
1104  Utot 2879033
1105  Utot 2680378
1106  Utot 2616883
1107  Utot 2702349
1108  Utot 2681738
1109  Utot 2708316
1110  Utot 2651769
1111  Utot 2697358
1112  Utot 2800333
1113  Utot 2589424
1114  Utot 2661461
1115  Utot 2611363
1116  Utot 2706124
1117  Utot 2814099
1118  Utot 2669157
1119  Utot 2760800
1120  Utot 2802714
1121  Utot 2697144
1122  Utot 2697916
1123  Utot 2784447
1124  Utot 2791818
1125  Utot 2567921
1126  Utot 2553930
1127  Utot 2736185
1128  Utot 2853658
1129  Utot 2825334
1130  Utot 2824599
1131  Utot 2822549
1132  Utot 2753195
1133  Utot 2619143
1134  Utot 2861412
1135  Utot 2621632
1136  Utot 2648346
1137  Utot 2803474
1138  Utot 2808799
1139  Utot 2731673
1140  Utot 2796550
1141  Utot 2854289
1142  Utot 2779750
1143  Utot 2785244
1144  Utot 2671388
1145  Utot 2679793
1146  Utot 2693586
1147  Utot 2672576
1148  Utot 2572949
1149  Utot 2566299
1150  Utot 2566708
1151  Utot 2761309
1152  Utot 2860966
1153  Utot 2765523
1154  Utot 2655441
1155  Utot 2856776
1156  Utot 2642410
1157  Utot 2484048
1158  Utot 2982920
1159  Utot 2498278
1160  Utot 2666906
1161  Utot 2579788
1162  Utot 2734207
1163  Utot 2794100
1164  Utot 2840349
1165  Utot 2649073
1166  Utot 2879488
1167  Utot 2861743
1168  Utot 2755762
1169  Utot 2671274
1170  Utot 2677253
1171  Utot 2676568
1172  Utot 2760474
1173  Utot 2646417
1174  Utot 2642596
1175  Utot 2726648
1176  Utot 2886577
1177  Utot 2694508
1178  Utot 2693354
1179  Utot 2755884
1180  Utot 2788087
1181  Utot 2771991
1182  Utot 2561531
1183  Utot 2707972
1184  Utot 2601102
1185  Utot 2718725
1186  Utot 2882122
1187  Utot 2527790
1188  Utot 2684463
1189  Utot 2796497
1190  Utot 2726078
1191  Utot 2729050
1192  Utot 2828951
1193  Utot 2671679
1194  Utot 2672071
1195  Utot 2860630
1196  Utot 2675761
1197  Utot 2547284
1198  Utot 2688398
1199  Utot 2648300
1200  Utot 2960115
1201  Utot 2864113
1202  Utot 2797527
1203  Utot 2594692
1204  Utot 2852663
1205  Utot 2821048
1206  Utot 2522273
1207  Utot 2663676
1208  Utot 2781330
1209  Utot 2835844
1210  Utot 2819631
1211  Utot 2729639
1212  Utot 2706519
1213  Utot 2693895
1214  Utot 2772780
1215  Utot 2589077
1216  Utot 2777358
1217  Utot 2562966
1218  Utot 2699390
1219  Utot 2635368
1220  Utot 2564012
1221  Utot 2882331
1222  Utot 2728383
1223  Utot 2699236
1224  Utot 2665235
1225  Utot 2823614
1226  Utot 2802660
1227  Utot 2784651
1228  Utot 2847771
1229  Utot 2809663
1230  Utot 2878179
1231  Utot 2683148
1232  Utot 2776949
1233  Utot 2923459
1234  Utot 2683320
1235  Utot 2860668
1236  Utot 2650588
1237  Utot 2725559
1238  Utot 2619675
1239  Utot 2794003
1240  Utot 2676633
1241  Utot 2625481
1242  Utot 2746646
1243  Utot 2890844
1244  Utot 2727224
1245  Utot 2569032
1246  Utot 2768491
1247  Utot 2626163
1248  Utot 2649438
1249  Utot 2847159
1250  Utot 2633391
1251  Utot 2599173
1252  Utot 2757536
1253  Utot 2739728
1254  Utot 2447872
1255  Utot 2541370
1256  Utot 2698064
1257  Utot 2712501
1258  Utot 2746210
1259  Utot 2734959
1260  Utot 2516979
1261  Utot 2774912
1262  Utot 2681588
1263  Utot 2701270
1264  Utot 2642403
1265  Utot 2700008
1266  Utot 2895085
1267  Utot 2770360
1268  Utot 2583325
1269  Utot 2814066
1270  Utot 2654499
1271  Utot 2734476
1272  Utot 2768032
1273  Utot 2843354
1274  Utot 2969948
1275  Utot 2885001
1276  Utot 2714438
1277  Utot 2710528
1278  Utot 2718608
1279  Utot 2805583
1280  Utot 2529704
1281  Utot 2822319
1282  Utot 2749519
1283  Utot 2698832
1284  Utot 2820301
1285  Utot 2574938
1286  Utot 2657639
1287  Utot 2645971
1288  Utot 2730090
1289  Utot 2680437
1290  Utot 2699911
1291  Utot 2901624
1292  Utot 2674366
1293  Utot 2589982
1294  Utot 2765038
1295  Utot 2614179
1296  Utot 2502612
1297  Utot 2715141
1298  Utot 2690040
1299  Utot 2774571
1300  Utot 2826367
1301  Utot 2709611
1302  Utot 2783392
1303  Utot 2732913
1304  Utot 2612179
1305  Utot 2848566
1306  Utot 2850001
1307  Utot 2748122
1308  Utot 2697786
1309  Utot 2517011
1310  Utot 2633917
1311  Utot 2533295
1312  Utot 2759204
1313  Utot 2748778
1314  Utot 2731398
1315  Utot 2768623
1316  Utot 2751762
1317  Utot 2643247
1318  Utot 2709318
1319  Utot 2822330
1320  Utot 2619045
1321  Utot 2829759
1322  Utot 2678465
1323  Utot 2572754
1324  Utot 2833800
1325  Utot 2712519
1326  Utot 2749998
1327  Utot 2737609
1328  Utot 2909414
1329  Utot 2616230
1330  Utot 2737961
1331  Utot 2904956
1332  Utot 2848080
1333  Utot 2759412
1334  Utot 2654847
1335  Utot 2629761
1336  Utot 2716412
1337  Utot 2761028
1338  Utot 2640830
1339  Utot 2687599
1340  Utot 2711156
1341  Utot 2759107
1342  Utot 2797330
1343  Utot 2723709
1344  Utot 2791002
1345  Utot 2736473
1346  Utot 2533291
1347  Utot 2736725
1348  Utot 2726514
1349  Utot 2692668
1350  Utot 2653469
1351  Utot 2738486
1352  Utot 2660038
1353  Utot 2711538
1354  Utot 2663420
1355  Utot 2702513
1356  Utot 2782729
1357  Utot 2910143
1358  Utot 2701917
1359  Utot 2768613
1360  Utot 2681231
1361  Utot 2645869
1362  Utot 2640981
1363  Utot 2389651
1364  Utot 2858249
1365  Utot 2731575
1366  Utot 2578556
1367  Utot 2735423
1368  Utot 2683964
1369  Utot 2758507
1370  Utot 2753115
1371  Utot 2703976
1372  Utot 2847807
1373  Utot 2790198
1374  Utot 2765709
1375  Utot 2702315
1376  Utot 2930540
1377  Utot 2744489
1378  Utot 2833607
1379  Utot 2956830
1380  Utot 2756526
1381  Utot 2601802
1382  Utot 2573561
1383  Utot 2626927
1384  Utot 3037060
1385  Utot 2791004
1386  Utot 2616883
1387  Utot 2572597
1388  Utot 2813842
1389  Utot 2650666
1390  Utot 2852644
1391  Utot 2613794
1392  Utot 2707445
1393  Utot 2644964
1394  Utot 2794656
1395  Utot 3101603
1396  Utot 2711153
1397  Utot 2889634
1398  Utot 2679920
1399  Utot 2729094
1400  Utot 2757651
1401  Utot 2776005
1402  Utot 2716015
1403  Utot 2725444
1404  Utot 2709070
1405  Utot 2671975
1406  Utot 2699077
1407  Utot 2589194
1408  Utot 2910277
1409  Utot 2691390
1410  Utot 2559677
1411  Utot 2621803
1412  Utot 2683984
1413  Utot 2675155
1414  Utot 2712232
1415  Utot 2643903
1416  Utot 2708605
1417  Utot 2736018
1418  Utot 2724708
1419  Utot 2730726
1420  Utot 2710453
1421  Utot 2757003
1422  Utot 2616363
1423  Utot 2670249
1424  Utot 2675081
1425  Utot 2692561
1426  Utot 2748827
1427  Utot 2660624
1428  Utot 2609929
1429  Utot 2680679
1430  Utot 2781769
1431  Utot 2623594
1432  Utot 2768360
1433  Utot 2769849
1434  Utot 2713819
1435  Utot 2771275
1436  Utot 2685018
1437  Utot 2763254
1438  Utot 2611786
1439  Utot 2556039
1440  Utot 2666782
1441  Utot 2749918
1442  Utot 2652510
1443  Utot 2699961
1444  Utot 2667917
1445  Utot 2703314
1446  Utot 2699212
1447  Utot 2696541
1448  Utot 2947295
1449  Utot 2628995
1450  Utot 2775040
1451  Utot 2776289
1452  Utot 2809517
1453  Utot 2636971
1454  Utot 2674182
1455  Utot 2771956
1456  Utot 2735185
1457  Utot 2732219
1458  Utot 2575494
1459  Utot 2680598
1460  Utot 2724722
1461  Utot 2611851
1462  Utot 2608154
1463  Utot 2747334
1464  Utot 2715143
1465  Utot 2841535
1466  Utot 2727142
1467  Utot 2853284
1468  Utot 2833242
1469  Utot 2594636
1470  Utot 2876316
1471  Utot 2772278
1472  Utot 2653611
1473  Utot 2676026
1474  Utot 2803461
1475  Utot 2703552
1476  Utot 2780094
1477  Utot 2668077
1478  Utot 2618831
1479  Utot 2782523
1480  Utot 2779751
1481  Utot 2524955
1482  Utot 2810648
1483  Utot 2716226
1484  Utot 2753664
1485  Utot 2668324
1486  Utot 2683354
1487  Utot 2648993
1488  Utot 2778556
1489  Utot 2745440
1490  Utot 2843569
1491  Utot 2759316
1492  Utot 2663548
1493  Utot 2748276
1494  Utot 2606812
1495  Utot 2774526
1496  Utot 2672921
1497  Utot 2680534
1498  Utot 2774114
1499  Utot 2650267
1500  Utot 2695650
1501  Utot 2680660
1502  Utot 2553337
1503  Utot 2730033
1504  Utot 2729534
1505  Utot 2682071
1506  Utot 2945241
1507  Utot 2773416
1508  Utot 2760178
1509  Utot 2758483
1510  Utot 2711416
1511  Utot 2629579
1512  Utot 2500501
1513  Utot 2610700
1514  Utot 2894469
1515  Utot 2671683
1516  Utot 2747213
1517  Utot 2839895
1518  Utot 2777614
1519  Utot 2629341
1520  Utot 2663151
1521  Utot 2787212
1522  Utot 2572436
1523  Utot 2751313
1524  Utot 2863662
1525  Utot 2911238
1526  Utot 2531116
1527  Utot 2629151
1528  Utot 2747015
1529  Utot 2906253
1530  Utot 2703041
1531  Utot 2674498
1532  Utot 2962241
1533  Utot 2633038
1534  Utot 2728204
1535  Utot 2598234
1536  Utot 2628579
1537  Utot 3009772
1538  Utot 2804393
1539  Utot 2604599
1540  Utot 2578533
1541  Utot 2781240
1542  Utot 2718145
1543  Utot 2751242
1544  Utot 2632138
1545  Utot 2554579
1546  Utot 2639434
1547  Utot 2783704
1548  Utot 2744540
1549  Utot 2836119
1550  Utot 2998366
1551  Utot 2867746
1552  Utot 2679020
1553  Utot 2723492
1554  Utot 2641747
1555  Utot 2819355
1556  Utot 2910906
1557  Utot 2498691
1558  Utot 2755892
1559  Utot 2685973
1560  Utot 2644928
1561  Utot 2570537
1562  Utot 2854226
1563  Utot 2939295
1564  Utot 2828663
1565  Utot 2749502
1566  Utot 2828399
1567  Utot 2721734
1568  Utot 2513473
1569  Utot 2649024
1570  Utot 2511528
1571  Utot 2655852
1572  Utot 2664146
1573  Utot 2606301
1574  Utot 2908433
1575  Utot 2765314
1576  Utot 2815222
1577  Utot 2845649
1578  Utot 2691833
1579  Utot 2796566
1580  Utot 2599650
1581  Utot 2610228
1582  Utot 2766635
1583  Utot 2682130
1584  Utot 2699205
1585  Utot 2555242
1586  Utot 2715513
1587  Utot 2781164
1588  Utot 2592845
1589  Utot 2692782
1590  Utot 2823479
1591  Utot 2741073
1592  Utot 2837625
1593  Utot 2909557
1594  Utot 2613788
1595  Utot 2776452
1596  Utot 2644169
1597  Utot 2711153
1598  Utot 2641683
1599  Utot 2801062
1600  Utot 2993621
1601  Utot 2591001
1602  Utot 2653930
1603  Utot 2632677
1604  Utot 2531391
1605  Utot 2664199
1606  Utot 2693084
1607  Utot 2713631
1608  Utot 2722273
1609  Utot 2755456
1610  Utot 2811611
1611  Utot 2770482
1612  Utot 2520313
1613  Utot 2641281
1614  Utot 2620716
1615  Utot 2564233
1616  Utot 2458695
1617  Utot 2946641
1618  Utot 2738446
1619  Utot 2718660
1620  Utot 2831836
1621  Utot 2663533
1622  Utot 2605974
1623  Utot 2684080
1624  Utot 2751305
1625  Utot 2659870
1626  Utot 2575451
1627  Utot 2761707
1628  Utot 2700873
1629  Utot 2954476
1630  Utot 2832939
1631  Utot 2625844
1632  Utot 2714497
1633  Utot 2710313
1634  Utot 2934566
1635  Utot 2800004
1636  Utot 2576912
1637  Utot 2715317
1638  Utot 2850789
1639  Utot 2672866
1640  Utot 2566278
1641  Utot 2805910
1642  Utot 2585412
1643  Utot 2537554
1644  Utot 2739513
1645  Utot 2837200
1646  Utot 2786017
1647  Utot 2669354
1648  Utot 2793109
1649  Utot 2881405
1650  Utot 2500405
1651  Utot 2667648
1652  Utot 2801891
1653  Utot 2640043
1654  Utot 2753281
1655  Utot 2782892
1656  Utot 2933772
1657  Utot 2776186
1658  Utot 2688982
1659  Utot 2908287
1660  Utot 2874343
1661  Utot 2794246
1662  Utot 2909658
1663  Utot 2697865
1664  Utot 2766284
1665  Utot 2670462
1666  Utot 2653684
1667  Utot 2559489
1668  Utot 2569431
1669  Utot 2741758
1670  Utot 2736855
1671  Utot 2836102
1672  Utot 2769806
1673  Utot 2696758
1674  Utot 2811447
1675  Utot 2690707
1676  Utot 2806887
1677  Utot 2683600
1678  Utot 2574376
1679  Utot 2787573
1680  Utot 2789589
1681  Utot 2716612
1682  Utot 2754070
1683  Utot 2844339
1684  Utot 2692575
1685  Utot 2806811
1686  Utot 2636655
1687  Utot 2782008
1688  Utot 2695010
1689  Utot 2594214
1690  Utot 2790097
1691  Utot 2817388
1692  Utot 2618558
1693  Utot 2795000
1694  Utot 2896239
1695  Utot 2723578
1696  Utot 2716016
1697  Utot 2716912
1698  Utot 2945536
1699  Utot 2752031
1700  Utot 2437306
1701  Utot 2843376
1702  Utot 2884751
1703  Utot 2605394
1704  Utot 2629481
1705  Utot 2578346
1706  Utot 2737036
1707  Utot 2887282
1708  Utot 2858476
1709  Utot 2636782
1710  Utot 2718250
1711  Utot 2683569
1712  Utot 2487812
1713  Utot 2611931
1714  Utot 2595328
1715  Utot 2693095
1716  Utot 2829258
1717  Utot 2523665
1718  Utot 2663074
1719  Utot 2841848
1720  Utot 2869253
1721  Utot 2711843
1722  Utot 2788557
1723  Utot 2681930
1724  Utot 2743825
1725  Utot 2565601
1726  Utot 2717773
1727  Utot 2749776
1728  Utot 2664384
1729  Utot 2752240
1730  Utot 2636332
1731  Utot 2884302
1732  Utot 2749275
1733  Utot 2800360
1734  Utot 2746395
1735  Utot 2780902
1736  Utot 2772717
1737  Utot 2653113
1738  Utot 2770142
1739  Utot 2552497
1740  Utot 2666634
1741  Utot 2729409
1742  Utot 2702461
1743  Utot 2743738
1744  Utot 2729636
1745  Utot 2714248
1746  Utot 2697568
1747  Utot 2678019
1748  Utot 2789026
1749  Utot 2734263
1750  Utot 2789426
1751  Utot 2682675
1752  Utot 2660709
1753  Utot 2578947
1754  Utot 2813924
1755  Utot 2844703
1756  Utot 2597286
1757  Utot 2635758
1758  Utot 2739452
1759  Utot 2779755
1760  Utot 2655328
1761  Utot 2859485
1762  Utot 2862108
1763  Utot 2610686
1764  Utot 2746785
1765  Utot 2682447
1766  Utot 2716657
1767  Utot 2612928
1768  Utot 2618021
1769  Utot 2511231
1770  Utot 2604866
1771  Utot 2914113
1772  Utot 2789269
1773  Utot 2636884
1774  Utot 2674174
1775  Utot 2759610
1776  Utot 2643381
1777  Utot 2791936
1778  Utot 2634035
1779  Utot 2915627
1780  Utot 2849606
1781  Utot 2785549
1782  Utot 2845017
1783  Utot 2736770
1784  Utot 2845219
1785  Utot 2686325
1786  Utot 2804350
1787  Utot 2658959
1788  Utot 2644622
1789  Utot 2652758
1790  Utot 2571552
1791  Utot 2811673
1792  Utot 2714296
1793  Utot 2675044
1794  Utot 2716344
1795  Utot 2761917
1796  Utot 2549079
1797  Utot 2686105
1798  Utot 2562072
1799  Utot 2571748
1800  Utot 2642144
1801  Utot 2646580
1802  Utot 2641923
1803  Utot 2660298
1804  Utot 2679886
1805  Utot 2621781
1806  Utot 2814883
1807  Utot 2707216
1808  Utot 2670873
1809  Utot 2740217
1810  Utot 2646420
1811  Utot 2600724
1812  Utot 2609570
1813  Utot 2905255
1814  Utot 2759255
1815  Utot 2616924
1816  Utot 2557798
1817  Utot 2720589
1818  Utot 2758976
1819  Utot 2656530
1820  Utot 2720216
1821  Utot 2821877
1822  Utot 2565219
1823  Utot 2500885
1824  Utot 2828940
1825  Utot 2711854
1826  Utot 2655415
1827  Utot 2805182
1828  Utot 2725663
1829  Utot 2693375
1830  Utot 2662636
1831  Utot 2693014
1832  Utot 3007542
1833  Utot 2674425
1834  Utot 2750687
1835  Utot 2816609
1836  Utot 2594412
1837  Utot 2679040
1838  Utot 2854454
1839  Utot 2618128
1840  Utot 2733112
1841  Utot 2754926
1842  Utot 2556736
1843  Utot 2939158
1844  Utot 2741680
1845  Utot 2653188
1846  Utot 2687671
1847  Utot 2674441
1848  Utot 2740106
1849  Utot 2683898
1850  Utot 2655605
1851  Utot 2767277
1852  Utot 2924219
1853  Utot 2679756
1854  Utot 2717430
1855  Utot 2749761
1856  Utot 2756332
1857  Utot 2729466
1858  Utot 2728913
1859  Utot 2857429
1860  Utot 2606062
1861  Utot 2823973
1862  Utot 2542353
1863  Utot 2707638
1864  Utot 2753117
1865  Utot 2826868
1866  Utot 2564222
1867  Utot 2761238
1868  Utot 2616217
1869  Utot 2569738
1870  Utot 2554043
1871  Utot 2575608
1872  Utot 2831033
1873  Utot 2573625
1874  Utot 2787128
1875  Utot 2871082
1876  Utot 2825511
1877  Utot 2678645
1878  Utot 2740136
1879  Utot 2791367
1880  Utot 2734092
1881  Utot 2646968
1882  Utot 2665062
1883  Utot 2726753
1884  Utot 2816517
1885  Utot 2538974
1886  Utot 2728535
1887  Utot 2580194
1888  Utot 2645241
1889  Utot 2670353
1890  Utot 2560745
1891  Utot 2846191
1892  Utot 2741914
1893  Utot 2636238
1894  Utot 2575106
1895  Utot 2851903
1896  Utot 2718687
1897  Utot 2690098
1898  Utot 2850972
1899  Utot 2849673
1900  Utot 2716342
1901  Utot 2725190
1902  Utot 2815922
1903  Utot 2790598
1904  Utot 2724719
1905  Utot 2728506
1906  Utot 2689294
1907  Utot 2870109
1908  Utot 2650174
1909  Utot 2699620
1910  Utot 2688370
1911  Utot 2725093
1912  Utot 2782565
1913  Utot 2711719
1914  Utot 2862479
1915  Utot 2636436
1916  Utot 2666805
1917  Utot 2835402
1918  Utot 2637899
1919  Utot 2648389
1920  Utot 2831154
1921  Utot 2696415
1922  Utot 2698216
1923  Utot 2555553
1924  Utot 2639891
1925  Utot 2604973
1926  Utot 2741001
1927  Utot 2676941
1928  Utot 2706446
1929  Utot 2783256
1930  Utot 2871178
1931  Utot 2755430
1932  Utot 2579214
1933  Utot 2905312
1934  Utot 2691451
1935  Utot 2933768
1936  Utot 2652010
1937  Utot 2532616
1938  Utot 2848383
1939  Utot 2635176
1940  Utot 2767918
1941  Utot 2874186
1942  Utot 2615063
1943  Utot 2673746
1944  Utot 2843736
1945  Utot 2656725
1946  Utot 2750187
1947  Utot 2648911
1948  Utot 2644750
1949  Utot 2671788
1950  Utot 2742223
1951  Utot 2772575
1952  Utot 2770755
1953  Utot 2787398
1954  Utot 2650310
1955  Utot 2620291
1956  Utot 3056794
1957  Utot 2563551
1958  Utot 2623431
1959  Utot 2864035
1960  Utot 2735055
1961  Utot 2736902
1962  Utot 2652644
1963  Utot 2741653
1964  Utot 2604568
1965  Utot 2789399
1966  Utot 2748088
1967  Utot 2529036
1968  Utot 2801127
1969  Utot 2709181
1970  Utot 2643391
1971  Utot 2671117
1972  Utot 2651378
1973  Utot 2715502
1974  Utot 2911756
1975  Utot 2700615
1976  Utot 2941183
1977  Utot 2702992
1978  Utot 2624649
1979  Utot 2634436
1980  Utot 2768680
1981  Utot 2690870
1982  Utot 2822514
1983  Utot 2666195
1984  Utot 2791015
1985  Utot 3042266
1986  Utot 2812080
1987  Utot 2619247
1988  Utot 2618086
1989  Utot 2706485
1990  Utot 2589216
1991  Utot 2820198
1992  Utot 2584964
1993  Utot 2564119
1994  Utot 2582746
1995  Utot 2757064
1996  Utot 2661302
1997  Utot 2854171
1998  Utot 2681309
1999  Utot 2817207
2000  Utot 2848094
2001  Utot 2668605
2002  Utot 2634757
2003  Utot 2817934
2004  Utot 2870783
2005  Utot 2595966
2006  Utot 2784124
2007  Utot 2595905
2008  Utot 2628936
2009  Utot 2798237
2010  Utot 2619998
2011  Utot 2767268
2012  Utot 2702249
2013  Utot 2712169
2014  Utot 2540839
2015  Utot 2709191
2016  Utot 2758570
2017  Utot 2568968
2018  Utot 2748978
2019  Utot 2622829
2020  Utot 2529340
2021  Utot 2713228
2022  Utot 2764860
2023  Utot 2628822
2024  Utot 2750688
2025  Utot 2677973
2026  Utot 2581024
2027  Utot 2744319
2028  Utot 2716369
2029  Utot 2723575
2030  Utot 2804569
2031  Utot 2929754
2032  Utot 2874607
2033  Utot 2677316
2034  Utot 2652633
2035  Utot 3074159
2036  Utot 2637920
2037  Utot 2733925
2038  Utot 2692529
2039  Utot 2668612
2040  Utot 2735525
2041  Utot 2605617
2042  Utot 2533534
2043  Utot 2642236
2044  Utot 2703661
2045  Utot 2953811
2046  Utot 2592783
2047  Utot 2655246
2048  Utot 2598602
2049  Utot 2692239
2050  Utot 2796200
2051  Utot 2831180
2052  Utot 2604991
2053  Utot 2801416
2054  Utot 2514753
2055  Utot 2685714
2056  Utot 2708464
2057  Utot 2555196
2058  Utot 2620249
2059  Utot 2594201
2060  Utot 2703761
2061  Utot 2865142
2062  Utot 2831270
2063  Utot 2700623
2064  Utot 2533871
2065  Utot 2793681
2066  Utot 2766637
2067  Utot 2958199
2068  Utot 2812864
2069  Utot 2744563
2070  Utot 2767518
2071  Utot 2566547
2072  Utot 2652553
2073  Utot 2710625
2074  Utot 2798974
2075  Utot 2726281
2076  Utot 2676549
2077  Utot 2849667
2078  Utot 2716990
2079  Utot 2703505
2080  Utot 2481169
2081  Utot 2507903
2082  Utot 2905720
2083  Utot 2622199
2084  Utot 2683098
2085  Utot 2555104
2086  Utot 2648093
2087  Utot 2785802
2088  Utot 2667010
2089  Utot 2642264
2090  Utot 2621734
2091  Utot 2661404
2092  Utot 2673627
2093  Utot 2591994
2094  Utot 2899894
2095  Utot 2678450
2096  Utot 2757666
2097  Utot 2845513
2098  Utot 2627585
2099  Utot 2609397
2100  Utot 2725542
2101  Utot 2858588
2102  Utot 2731461
2103  Utot 2671558
2104  Utot 2604908
2105  Utot 2806542
2106  Utot 2824846
2107  Utot 2788120
2108  Utot 2806719
2109  Utot 2748110
2110  Utot 2795465
2111  Utot 2579013
2112  Utot 2572743
2113  Utot 2637939
2114  Utot 2712953
2115  Utot 2736445
2116  Utot 2600479
2117  Utot 2706977
2118  Utot 2795664
2119  Utot 2591863
2120  Utot 2600552
2121  Utot 2754832
2122  Utot 2606482
2123  Utot 2675618
2124  Utot 2629515
2125  Utot 2584161
2126  Utot 2688842
2127  Utot 2763201
2128  Utot 2729604
2129  Utot 2650231
2130  Utot 2567382
2131  Utot 2731121
2132  Utot 2497591
2133  Utot 2689664
2134  Utot 2667384
2135  Utot 2625422
2136  Utot 2768132
2137  Utot 2886848
2138  Utot 2717615
2139  Utot 2952202
2140  Utot 2609366
2141  Utot 2657387
2142  Utot 2759904
2143  Utot 2840618
2144  Utot 2639545
2145  Utot 2821023
2146  Utot 2667490
2147  Utot 2844918
2148  Utot 2739274
2149  Utot 2738313
2150  Utot 2743837
2151  Utot 2775548
2152  Utot 2721266
2153  Utot 2650860
2154  Utot 2558957
2155  Utot 2847603
2156  Utot 2692949
2157  Utot 2818087
2158  Utot 2820770
2159  Utot 2949927
2160  Utot 2897114
2161  Utot 2778256
2162  Utot 2983835
2163  Utot 2575631
2164  Utot 2740113
2165  Utot 2606772
2166  Utot 2705658
2167  Utot 2813376
2168  Utot 2781340
2169  Utot 2592517
2170  Utot 2619402
2171  Utot 2809643
2172  Utot 2867526
2173  Utot 2872345
2174  Utot 2779385
2175  Utot 2876773
2176  Utot 2536869
2177  Utot 2622561
2178  Utot 2967036
2179  Utot 2738026
2180  Utot 2517509
2181  Utot 2653877
2182  Utot 2733731
2183  Utot 2728329
2184  Utot 2739528
2185  Utot 2524749
2186  Utot 2749248
2187  Utot 2525207
2188  Utot 2654240
2189  Utot 2643343
2190  Utot 2675290
2191  Utot 2693342
2192  Utot 2744487
2193  Utot 2885365
2194  Utot 2640619
2195  Utot 2557542
2196  Utot 2895407
2197  Utot 2667313
2198  Utot 2672186
2199  Utot 2632964
2200  Utot 2715920
2201  Utot 2587272
2202  Utot 2739563
2203  Utot 2708114
2204  Utot 2654514
2205  Utot 2683900
2206  Utot 2729902
2207  Utot 2587012
2208  Utot 2762713
2209  Utot 2903069
2210  Utot 2690056
2211  Utot 2881401
2212  Utot 2854886
2213  Utot 2814565
2214  Utot 2741426
2215  Utot 2627501
2216  Utot 2742821
2217  Utot 2658141
2218  Utot 2888536
2219  Utot 2697909
2220  Utot 2627362
2221  Utot 2697493
2222  Utot 2788092
2223  Utot 2787775
2224  Utot 2793008
2225  Utot 2814317
2226  Utot 2806419
2227  Utot 2698337
2228  Utot 2790558
2229  Utot 2823931
2230  Utot 2511388
2231  Utot 2538872
2232  Utot 2766877
2233  Utot 2645796
2234  Utot 2717142
2235  Utot 2672864
2236  Utot 2852797
2237  Utot 2687380
2238  Utot 2601798
2239  Utot 2535403
2240  Utot 2651812
2241  Utot 2584347
2242  Utot 2777666
2243  Utot 2685249
2244  Utot 2632154
2245  Utot 2797364
2246  Utot 2778156
2247  Utot 2816978
2248  Utot 2620807
2249  Utot 2762967
2250  Utot 2800822
2251  Utot 2606535
2252  Utot 2388322
2253  Utot 2676296
2254  Utot 2640688
2255  Utot 2730058
2256  Utot 2643200
2257  Utot 2850177
2258  Utot 2640647
2259  Utot 2785945
2260  Utot 2847582
2261  Utot 2668331
2262  Utot 2697474
2263  Utot 2743064
2264  Utot 2586363
2265  Utot 2679263
2266  Utot 2623268
2267  Utot 2874471
2268  Utot 2785456
2269  Utot 2915700
2270  Utot 2652769
2271  Utot 2709957
2272  Utot 2601138
2273  Utot 2862046
2274  Utot 2886374
2275  Utot 2562326
2276  Utot 2783806
2277  Utot 2839032
2278  Utot 2775269
2279  Utot 2780764
2280  Utot 2632741
2281  Utot 2600705
2282  Utot 2575929
2283  Utot 2733212
2284  Utot 2532028
2285  Utot 2868394
2286  Utot 2724944
2287  Utot 2714006
2288  Utot 2665169
2289  Utot 2606712
2290  Utot 2706541
2291  Utot 2846886
2292  Utot 3026557
2293  Utot 2596426
2294  Utot 2911183
2295  Utot 2567862
2296  Utot 2679213
2297  Utot 2700551
2298  Utot 2677322
2299  Utot 2900670
2300  Utot 2612738
2301  Utot 2840109
2302  Utot 2704271
2303  Utot 2738579
2304  Utot 2824526
2305  Utot 2609107
2306  Utot 2623448
2307  Utot 2765000
2308  Utot 2793433
2309  Utot 2712562
2310  Utot 2675122
2311  Utot 2992781
2312  Utot 2836735
2313  Utot 2790773
2314  Utot 2790058
2315  Utot 2572930
2316  Utot 2849347
2317  Utot 2702428
2318  Utot 2758444
2319  Utot 2783985
2320  Utot 2529246
2321  Utot 2507747
2322  Utot 2804787
2323  Utot 2854441
2324  Utot 2823417
2325  Utot 2782106
2326  Utot 2658891
2327  Utot 2650127
2328  Utot 2580670
2329  Utot 2748178
2330  Utot 2658156
2331  Utot 2749607
2332  Utot 2716384
2333  Utot 2688070
2334  Utot 2673886
2335  Utot 2965267
2336  Utot 2652761
2337  Utot 2555021
2338  Utot 2717578
2339  Utot 2768741
2340  Utot 2619342
2341  Utot 2646736
2342  Utot 2887680
2343  Utot 2678758
2344  Utot 2620950
2345  Utot 2761298
2346  Utot 2768840
2347  Utot 2733012
2348  Utot 2586263
2349  Utot 2645489
2350  Utot 2618985
2351  Utot 2700532
2352  Utot 2763961
2353  Utot 2747050
2354  Utot 2720257
2355  Utot 2775708
2356  Utot 2861110
2357  Utot 2690756
2358  Utot 2721066
2359  Utot 2625466
2360  Utot 2627384
2361  Utot 2739160
2362  Utot 2580727
2363  Utot 2691848
2364  Utot 2647465
2365  Utot 2779433
2366  Utot 2691391
2367  Utot 2740828
2368  Utot 2835045
2369  Utot 2863075
2370  Utot 2779002
2371  Utot 2663080
2372  Utot 2817170
2373  Utot 2706877
2374  Utot 2703716
2375  Utot 2685125
2376  Utot 2751803
2377  Utot 2674832
2378  Utot 2914303
2379  Utot 2654738
2380  Utot 2755223
2381  Utot 2882396
2382  Utot 2758707
2383  Utot 2696210
2384  Utot 2787818
2385  Utot 2778790
2386  Utot 2599223
2387  Utot 2678143
2388  Utot 2636270
2389  Utot 2653722
2390  Utot 2547172
2391  Utot 2672596
2392  Utot 2678364
2393  Utot 2578704
2394  Utot 2885880
2395  Utot 2691165
2396  Utot 2888530
2397  Utot 2730189
2398  Utot 2746429
2399  Utot 2802620
2400  Utot 2727410
2401  Utot 2780405
2402  Utot 2768444
2403  Utot 2799563
2404  Utot 2679547
2405  Utot 2786595
2406  Utot 2733150
2407  Utot 2874854
2408  Utot 2816972
2409  Utot 2630326
2410  Utot 2954665
2411  Utot 2746787
2412  Utot 2857446
2413  Utot 2608465
2414  Utot 2799308
2415  Utot 2523242
2416  Utot 2661204
2417  Utot 2698203
2418  Utot 2721619
2419  Utot 2768779
2420  Utot 2743806
2421  Utot 2779235
2422  Utot 2651117
2423  Utot 2561187
2424  Utot 2731286
2425  Utot 2693790
2426  Utot 2666409
2427  Utot 2802567
2428  Utot 2764144
2429  Utot 2722855
2430  Utot 2771478
2431  Utot 2608106
2432  Utot 2769923
2433  Utot 2618965
2434  Utot 2615350
2435  Utot 2732487
2436  Utot 2730044
2437  Utot 2539737
2438  Utot 2894850
2439  Utot 2736328
2440  Utot 2785613
2441  Utot 2483593
2442  Utot 2816599
2443  Utot 2441692
2444  Utot 2766703
2445  Utot 2773198
2446  Utot 2683575
2447  Utot 2731689
2448  Utot 2588074
2449  Utot 2699544
2450  Utot 2692808
2451  Utot 2917482
2452  Utot 2580806
2453  Utot 2742418
2454  Utot 2618875
2455  Utot 2619990
2456  Utot 2846977
2457  Utot 2940935
2458  Utot 2778213
2459  Utot 2787270
2460  Utot 2646994
2461  Utot 2571836
2462  Utot 2762082
2463  Utot 2805537
2464  Utot 2678165
2465  Utot 2995841
2466  Utot 2563230
2467  Utot 2797767
2468  Utot 2787970
2469  Utot 2576427
2470  Utot 2709344
2471  Utot 2689881
2472  Utot 2613630
2473  Utot 2713009
2474  Utot 2766959
2475  Utot 2698604
2476  Utot 2566839
2477  Utot 2705512
2478  Utot 2831226
2479  Utot 2892693
2480  Utot 2731789
2481  Utot 2659102
2482  Utot 2595254
2483  Utot 2859281
2484  Utot 2644746
2485  Utot 2661618
2486  Utot 2573152
2487  Utot 2695357
2488  Utot 2591076
2489  Utot 2650672
2490  Utot 2684322
2491  Utot 3034981
2492  Utot 2764615
2493  Utot 2682304
2494  Utot 2812895
2495  Utot 2600769
2496  Utot 2694041
2497  Utot 2886393
2498  Utot 2800144
2499  Utot 2780959
2500  Utot 2671246
2501  Utot 2623255
2502  Utot 2698177
2503  Utot 2801033
2504  Utot 2609393
2505  Utot 2804069
2506  Utot 2714114
2507  Utot 2733317
2508  Utot 2541450
2509  Utot 2777008
2510  Utot 2809334
2511  Utot 2835419
2512  Utot 2562444
2513  Utot 2725996
2514  Utot 2661421
2515  Utot 2709708
2516  Utot 2864272
2517  Utot 2941401
2518  Utot 2686392
2519  Utot 2650740
2520  Utot 2746667
2521  Utot 2567673
2522  Utot 2672044
2523  Utot 2728318
2524  Utot 2709813
2525  Utot 2676451
2526  Utot 2573334
2527  Utot 2539195
2528  Utot 2625506
2529  Utot 2899930
2530  Utot 2657016
2531  Utot 2698606
2532  Utot 2949088
2533  Utot 2617557
2534  Utot 2736963
2535  Utot 2766554
2536  Utot 2723747
2537  Utot 2791480
2538  Utot 2722524
2539  Utot 2758820
2540  Utot 2642888
2541  Utot 2740356
2542  Utot 2707167
2543  Utot 2887974
2544  Utot 2787113
2545  Utot 2719464
2546  Utot 2815437
2547  Utot 2751105
2548  Utot 2700975
2549  Utot 2621514
2550  Utot 2629273
2551  Utot 2804782
2552  Utot 2775784
2553  Utot 2721673
2554  Utot 2577363
2555  Utot 2576187
2556  Utot 2777572
2557  Utot 2707831
2558  Utot 2842877
2559  Utot 2754025
2560  Utot 2512592
2561  Utot 2699846
2562  Utot 2830229
2563  Utot 2827523
2564  Utot 2690075
2565  Utot 2609238
2566  Utot 2652456
2567  Utot 2734907
2568  Utot 2715079
2569  Utot 2848973
2570  Utot 2752515
2571  Utot 2710304
2572  Utot 2686731
2573  Utot 2648103
2574  Utot 2807122
2575  Utot 2749641
2576  Utot 2574155
2577  Utot 2780754
2578  Utot 2691606
2579  Utot 2930151
2580  Utot 2830032
2581  Utot 2788094
2582  Utot 2645985
2583  Utot 3096051
2584  Utot 2687896
2585  Utot 2712324
2586  Utot 2662745
2587  Utot 2560993
2588  Utot 2698302
2589  Utot 2647209
2590  Utot 2737804
2591  Utot 2817249
2592  Utot 2579154
2593  Utot 2736193
2594  Utot 2700872
2595  Utot 2580887
2596  Utot 2630470
2597  Utot 2830934
2598  Utot 2915103
2599  Utot 2932232
2600  Utot 2730206
2601  Utot 2709054
2602  Utot 2829837
2603  Utot 2927568
2604  Utot 2572941
2605  Utot 2690154
2606  Utot 2710020
2607  Utot 2713116
2608  Utot 2621638
2609  Utot 2945856
2610  Utot 2731114
2611  Utot 2675586
2612  Utot 2681476
2613  Utot 2610581
2614  Utot 2764469
2615  Utot 2714853
2616  Utot 2707127
2617  Utot 2701748
2618  Utot 2736266
2619  Utot 2552412
2620  Utot 2768724
2621  Utot 2652480
2622  Utot 2780499
2623  Utot 2899674
2624  Utot 2790229
2625  Utot 2507449
2626  Utot 2807633
2627  Utot 2619353
2628  Utot 2749412
2629  Utot 2628251
2630  Utot 2761743
2631  Utot 2635115
2632  Utot 2945521
2633  Utot 2798230
2634  Utot 2585796
2635  Utot 2448000
2636  Utot 2721496
2637  Utot 2766360
2638  Utot 2747817
2639  Utot 2605962
2640  Utot 2817938
2641  Utot 2577162
2642  Utot 2826988
2643  Utot 2663442
2644  Utot 2428243
2645  Utot 2824548
2646  Utot 2646663
2647  Utot 2763492
2648  Utot 2798860
2649  Utot 2751053
2650  Utot 2612195
2651  Utot 2724069
2652  Utot 2808416
2653  Utot 2732508
2654  Utot 2658123
2655  Utot 2718300
2656  Utot 2636128
2657  Utot 2644849
2658  Utot 2681276
2659  Utot 2818445
2660  Utot 2779362
2661  Utot 2699102
2662  Utot 2679214
2663  Utot 2802588
2664  Utot 2769134
2665  Utot 2682310
2666  Utot 3011742
2667  Utot 2678189
2668  Utot 2774770
2669  Utot 2782400
2670  Utot 2608312
2671  Utot 2717638
2672  Utot 2841990
2673  Utot 2886372
2674  Utot 2644015
2675  Utot 2549632
2676  Utot 2766218
2677  Utot 2750198
2678  Utot 2672989
2679  Utot 2761502
2680  Utot 2628897
2681  Utot 2664053
2682  Utot 2611645
2683  Utot 2589766
2684  Utot 2764507
2685  Utot 2698256
2686  Utot 2754679
2687  Utot 2667282
2688  Utot 2630867
2689  Utot 2875748
2690  Utot 2820728
2691  Utot 2710840
2692  Utot 2572802
2693  Utot 2695641
2694  Utot 2711905
2695  Utot 2739882
2696  Utot 2632150
2697  Utot 2863368
2698  Utot 2595405
2699  Utot 2682004
2700  Utot 2664531
2701  Utot 2660053
2702  Utot 2626868
2703  Utot 2692696
2704  Utot 2627044
2705  Utot 2968514
2706  Utot 2680492
2707  Utot 2698975
2708  Utot 2922516
2709  Utot 2783548
2710  Utot 2801566
2711  Utot 2804730
2712  Utot 2743442
2713  Utot 2858555
2714  Utot 2731500
2715  Utot 2644524
2716  Utot 2600018
2717  Utot 2516123
2718  Utot 2610950
2719  Utot 2702523
2720  Utot 2679369
2721  Utot 2705718
2722  Utot 2720157
2723  Utot 2692907
2724  Utot 2615475
2725  Utot 2719799
2726  Utot 2664966
2727  Utot 2832434
2728  Utot 2738777
2729  Utot 2876298
2730  Utot 2683514
2731  Utot 2733699
2732  Utot 2796591
2733  Utot 2737230
2734  Utot 2617061
2735  Utot 2699730
2736  Utot 2607766
2737  Utot 2813697
2738  Utot 2630410
2739  Utot 2701398
2740  Utot 2838653
2741  Utot 2721339
2742  Utot 2651987
2743  Utot 2650940
2744  Utot 2762216
2745  Utot 2651682
2746  Utot 2644911
2747  Utot 2615048
2748  Utot 2896962
2749  Utot 2614662
2750  Utot 2717394
2751  Utot 2786628
2752  Utot 2764300
2753  Utot 2597568
2754  Utot 2712244
2755  Utot 2394822
2756  Utot 2515389
2757  Utot 2722333
2758  Utot 2616312
2759  Utot 2784470
2760  Utot 2693789
2761  Utot 2797085
2762  Utot 2869965
2763  Utot 2956841
2764  Utot 2661196
2765  Utot 2821203
2766  Utot 2681809
2767  Utot 2586266
2768  Utot 2747297
2769  Utot 2798071
2770  Utot 2741234
2771  Utot 2650804
2772  Utot 2667804
2773  Utot 2777259
2774  Utot 2749152
2775  Utot 2653005
2776  Utot 2604081
2777  Utot 2614645
2778  Utot 2663692
2779  Utot 2809135
2780  Utot 2859786
2781  Utot 2731008
2782  Utot 2699118
2783  Utot 2631612
2784  Utot 2660085
2785  Utot 2638924
2786  Utot 2795686
2787  Utot 2709965
2788  Utot 2869809
2789  Utot 2733016
2790  Utot 2755204
2791  Utot 2894313
2792  Utot 2846518
2793  Utot 2621510
2794  Utot 2752302
2795  Utot 2810516
2796  Utot 2825499
2797  Utot 2923880
2798  Utot 2727782
2799  Utot 2716708
2800  Utot 2745573
2801  Utot 2696126
2802  Utot 2740982
2803  Utot 2555606
2804  Utot 2604109
2805  Utot 2670821
2806  Utot 2724977
2807  Utot 2857287
2808  Utot 2590389
2809  Utot 2794412
2810  Utot 2602079
2811  Utot 2508210
2812  Utot 2773178
2813  Utot 2553302
2814  Utot 2610640
2815  Utot 2754148
2816  Utot 2643599
2817  Utot 2657427
2818  Utot 2787599
2819  Utot 2858795
2820  Utot 2597518
2821  Utot 2460321
2822  Utot 2687053
2823  Utot 2844183
2824  Utot 2572194
2825  Utot 2597350
2826  Utot 2712802
2827  Utot 2782765
2828  Utot 2622689
2829  Utot 2829836
2830  Utot 2584626
2831  Utot 2719006
2832  Utot 2733052
2833  Utot 2552368
2834  Utot 2842510
2835  Utot 2586651
2836  Utot 2815583
2837  Utot 2745339
2838  Utot 2616628
2839  Utot 2735052
2840  Utot 2621168
2841  Utot 2650218
2842  Utot 2692022
2843  Utot 2760314
2844  Utot 2783801
2845  Utot 2577660
2846  Utot 2785914
2847  Utot 2523687
2848  Utot 2801738
2849  Utot 2685105
2850  Utot 2896043
2851  Utot 2982811
2852  Utot 2593275
2853  Utot 2610159
2854  Utot 2964016
2855  Utot 2806650
2856  Utot 2845913
2857  Utot 2657374
2858  Utot 2684051
2859  Utot 2656638
2860  Utot 2599302
2861  Utot 2714120
2862  Utot 2569071
2863  Utot 2969230
2864  Utot 2677156
2865  Utot 2669042
2866  Utot 2742956
2867  Utot 2565800
2868  Utot 2578241
2869  Utot 2703569
2870  Utot 2734995
2871  Utot 2733780
2872  Utot 2885243
2873  Utot 2779641
2874  Utot 2754601
2875  Utot 2901418
2876  Utot 2875602
2877  Utot 2590917
2878  Utot 2711035
2879  Utot 2725841
2880  Utot 2547118
2881  Utot 2743278
2882  Utot 2769362
2883  Utot 2547225
2884  Utot 2743237
2885  Utot 2729053
2886  Utot 2599452
2887  Utot 2662723
2888  Utot 2692061
2889  Utot 2767304
2890  Utot 2853381
2891  Utot 2635098
2892  Utot 2659529
2893  Utot 2457089
2894  Utot 2646849
2895  Utot 2612190
2896  Utot 2684506
2897  Utot 2848012
2898  Utot 2719164
2899  Utot 2656923
2900  Utot 2641076
2901  Utot 2893863
2902  Utot 2608787
2903  Utot 2873110
2904  Utot 2938445
2905  Utot 2841701
2906  Utot 2663358
2907  Utot 2849233
2908  Utot 2727558
2909  Utot 2519531
2910  Utot 2749239
2911  Utot 2744162
2912  Utot 2860706
2913  Utot 2706546
2914  Utot 2678381
2915  Utot 2895181
2916  Utot 2783861
2917  Utot 2652765
2918  Utot 2788885
2919  Utot 2581403
2920  Utot 2707447
2921  Utot 2787826
2922  Utot 2873174
2923  Utot 2687988
2924  Utot 2769713
2925  Utot 2760637
2926  Utot 2841267
2927  Utot 2768867
2928  Utot 2705415
2929  Utot 2646496
2930  Utot 2758122
2931  Utot 2687333
2932  Utot 2705706
2933  Utot 2872405
2934  Utot 2805774
2935  Utot 2641874
2936  Utot 2731806
2937  Utot 2721801
2938  Utot 2815122
2939  Utot 2799708
2940  Utot 2590898
2941  Utot 2801096
2942  Utot 2800610
2943  Utot 2679589
2944  Utot 2788510
2945  Utot 2851151
2946  Utot 2657355
2947  Utot 2483495
2948  Utot 2704468
2949  Utot 2553151
2950  Utot 2550685
2951  Utot 2723282
2952  Utot 2740933
2953  Utot 2865062
2954  Utot 2748334
2955  Utot 2680067
2956  Utot 2768071
2957  Utot 2653621
2958  Utot 2632921
2959  Utot 2750088
2960  Utot 2815414
2961  Utot 2803926
2962  Utot 2719036
2963  Utot 2912713
2964  Utot 2718632
2965  Utot 2839846
2966  Utot 2773046
2967  Utot 2813581
2968  Utot 2697711
2969  Utot 2742761
2970  Utot 2790785
2971  Utot 2553992
2972  Utot 2930881
2973  Utot 2687085
2974  Utot 2861549
2975  Utot 2786011
2976  Utot 2954538
2977  Utot 2889810
2978  Utot 2640973
2979  Utot 2645288
2980  Utot 2686641
2981  Utot 2628777
2982  Utot 2683785
2983  Utot 2730237
2984  Utot 2591527
2985  Utot 2743887
2986  Utot 2662150
2987  Utot 2840066
2988  Utot 2781882
2989  Utot 2761507
2990  Utot 2746119
2991  Utot 2746787
2992  Utot 2681817
2993  Utot 2633672
2994  Utot 2819073
2995  Utot 2658141
2996  Utot 2807684
2997  Utot 2692705
2998  Utot 2821619
2999  Utot 2685566
3000  Utot 2735932
3001  Utot 2746325
3002  Utot 2740822
3003  Utot 2733416
3004  Utot 2531167
3005  Utot 2790488
3006  Utot 2858706
3007  Utot 2706798
3008  Utot 2703827
3009  Utot 2468342
3010  Utot 2918679
3011  Utot 2581338
3012  Utot 2814906
3013  Utot 2772076
3014  Utot 2749885
3015  Utot 2727011
3016  Utot 2899860
3017  Utot 2890977
3018  Utot 2787309
3019  Utot 2822968
3020  Utot 2756690
3021  Utot 2790895
3022  Utot 2905348
3023  Utot 2536556
3024  Utot 2787684
3025  Utot 2815189
3026  Utot 2677964
3027  Utot 2864106
3028  Utot 2618930
3029  Utot 2619374
3030  Utot 2725622
3031  Utot 2748003
3032  Utot 2650582
3033  Utot 2683559
3034  Utot 2637835
3035  Utot 2869169
3036  Utot 2749762
3037  Utot 2723863
3038  Utot 2664365
3039  Utot 2676252
3040  Utot 2755525
3041  Utot 2683537
3042  Utot 2841310
3043  Utot 2730750
3044  Utot 2859533
3045  Utot 2696491
3046  Utot 2764454
3047  Utot 2853433
3048  Utot 2691995
3049  Utot 2824152
3050  Utot 2709791
3051  Utot 2792498
3052  Utot 2669218
3053  Utot 2481563
3054  Utot 2815633
3055  Utot 2501784
3056  Utot 2845944
3057  Utot 2895173
3058  Utot 2706572
3059  Utot 2754715
3060  Utot 2728876
3061  Utot 2902632
3062  Utot 2833397
3063  Utot 2688646
3064  Utot 2666319
3065  Utot 2640942
3066  Utot 2582198
3067  Utot 2611344
3068  Utot 2753129
3069  Utot 2675645
3070  Utot 2871560
3071  Utot 2733478
3072  Utot 2614616
3073  Utot 2712443
3074  Utot 2570859
3075  Utot 2673838
3076  Utot 2723699
3077  Utot 2793679
3078  Utot 2805167
3079  Utot 2618303
3080  Utot 2805571
3081  Utot 2734214
3082  Utot 2716021
3083  Utot 2681652
3084  Utot 2631418
3085  Utot 2747996
3086  Utot 2618165
3087  Utot 2659287
3088  Utot 2811579
3089  Utot 2667203
3090  Utot 2611909
3091  Utot 2796743
3092  Utot 2778768
3093  Utot 2612666
3094  Utot 2720456
3095  Utot 2683354
3096  Utot 2756927
3097  Utot 2898093
3098  Utot 2702365
3099  Utot 2724569
3100  Utot 2749290
3101  Utot 2676514
3102  Utot 2729535
3103  Utot 2535387
3104  Utot 2727886
3105  Utot 2656324
3106  Utot 2893675
3107  Utot 2618974
3108  Utot 2770719
3109  Utot 2663226
3110  Utot 2820515
3111  Utot 2711795
3112  Utot 2637186
3113  Utot 2567548
3114  Utot 2679409
3115  Utot 2486999
3116  Utot 2715584
3117  Utot 2605169
3118  Utot 2589778
3119  Utot 2575277
3120  Utot 2645661
3121  Utot 2678330
3122  Utot 2883877
3123  Utot 2835624
3124  Utot 2630882
3125  Utot 2920653
3126  Utot 2612435
3127  Utot 2849253
3128  Utot 2541093
3129  Utot 2729563
3130  Utot 2560177
3131  Utot 2607731
3132  Utot 2691780
3133  Utot 2829194
3134  Utot 2678277
3135  Utot 2552832
3136  Utot 2811361
3137  Utot 2618736
3138  Utot 2767175
3139  Utot 2645629
3140  Utot 2762385
3141  Utot 2636250
3142  Utot 2743631
3143  Utot 2733498
3144  Utot 2695826
3145  Utot 2778933
3146  Utot 2691389
3147  Utot 2682479
3148  Utot 2743752
3149  Utot 2800666
3150  Utot 2794168
3151  Utot 2638189
3152  Utot 2641935
3153  Utot 2814411
3154  Utot 2569077
3155  Utot 2748827
3156  Utot 2796505
3157  Utot 2781311
3158  Utot 2712254
3159  Utot 2867769
3160  Utot 2727194
3161  Utot 2714166
3162  Utot 2643857
3163  Utot 2749619
3164  Utot 2882082
3165  Utot 2653437
3166  Utot 2663325
3167  Utot 2772397
3168  Utot 2957520
3169  Utot 2710200
3170  Utot 2671152
3171  Utot 2709594
3172  Utot 2710728
3173  Utot 2683311
3174  Utot 2764900
3175  Utot 2448238
3176  Utot 2760756
3177  Utot 2715216
3178  Utot 2570083
3179  Utot 2841201
3180  Utot 2680108
3181  Utot 2918481
3182  Utot 2749121
3183  Utot 2644281
3184  Utot 2852000
3185  Utot 2774000
3186  Utot 2679408
3187  Utot 2812947
3188  Utot 2697998
3189  Utot 2567913
3190  Utot 2843748
3191  Utot 2617192
3192  Utot 2568638
3193  Utot 2729225
3194  Utot 2706225
3195  Utot 2596708
3196  Utot 2637739
3197  Utot 2664835
3198  Utot 2691484
3199  Utot 2722051
3200  Utot 2784646
3201  Utot 2843596
3202  Utot 2750732
3203  Utot 2579625
3204  Utot 2611665
3205  Utot 2798487
3206  Utot 2724803
3207  Utot 2684917
3208  Utot 2574637
3209  Utot 2928169
3210  Utot 2571891
3211  Utot 2761950
3212  Utot 2956717
3213  Utot 2708360
3214  Utot 2974869
3215  Utot 2599392
3216  Utot 2547404
3217  Utot 2745703
3218  Utot 2802572
3219  Utot 2623249
3220  Utot 2746460
3221  Utot 2747057
3222  Utot 2903756
3223  Utot 2670492
3224  Utot 2808927
3225  Utot 2653836
3226  Utot 2844719
3227  Utot 2781058
3228  Utot 2560247
3229  Utot 2723070
3230  Utot 2740845
3231  Utot 2726392
3232  Utot 2838885
3233  Utot 2579648
3234  Utot 2529369
3235  Utot 2644978
3236  Utot 2706568
3237  Utot 2760322
3238  Utot 2663057
3239  Utot 2579436
3240  Utot 2634961
3241  Utot 3013917
3242  Utot 2709001
3243  Utot 2687106
3244  Utot 2703923
3245  Utot 2616003
3246  Utot 2730714
3247  Utot 2734292
3248  Utot 2726206
3249  Utot 2678949
3250  Utot 2826693
3251  Utot 2614040
3252  Utot 2642656
3253  Utot 2601291
3254  Utot 2795239
3255  Utot 2672921
3256  Utot 2746533
3257  Utot 2745180
3258  Utot 2747929
3259  Utot 2775328
3260  Utot 2840869
3261  Utot 2685592
3262  Utot 2645980
3263  Utot 2535063
3264  Utot 2771847
3265  Utot 2783850
3266  Utot 2679823
3267  Utot 2646074
3268  Utot 2803459
3269  Utot 2595381
3270  Utot 2748928
3271  Utot 2823625
3272  Utot 2898523
3273  Utot 3029687
3274  Utot 2773874
3275  Utot 2727224
3276  Utot 2734865
3277  Utot 2818077
3278  Utot 2762628
3279  Utot 2632905
3280  Utot 2797607
3281  Utot 2559308
3282  Utot 2563630
3283  Utot 2883845
3284  Utot 2852065
3285  Utot 2503738
3286  Utot 2728561
3287  Utot 2734351
3288  Utot 2859309
3289  Utot 2734176
3290  Utot 2552785
3291  Utot 2650273
3292  Utot 2649217
3293  Utot 2753166
3294  Utot 2700736
3295  Utot 2711575
3296  Utot 2748198
3297  Utot 2693566
3298  Utot 2769473
3299  Utot 2595343
3300  Utot 2581938
3301  Utot 2768064
3302  Utot 2611755
3303  Utot 2496737
3304  Utot 2605214
3305  Utot 2680529
3306  Utot 2782329
3307  Utot 2909167
3308  Utot 2562865
3309  Utot 2918814
3310  Utot 2730696
3311  Utot 3074014
3312  Utot 2636853
3313  Utot 2838147
3314  Utot 2598530
3315  Utot 2735084
3316  Utot 2454812
3317  Utot 2648922
3318  Utot 2577377
3319  Utot 2666227
3320  Utot 2803661
3321  Utot 2773039
3322  Utot 2870641
3323  Utot 2603810
3324  Utot 2762430
3325  Utot 2652160
3326  Utot 2599261
3327  Utot 2519821
3328  Utot 2719705
3329  Utot 2645922
3330  Utot 2729319
3331  Utot 2713263
3332  Utot 2791334
3333  Utot 2712444
3334  Utot 2766852
3335  Utot 2613631
3336  Utot 2596063
3337  Utot 2660579
3338  Utot 2787155
3339  Utot 2650982
3340  Utot 2671984
3341  Utot 2659147
3342  Utot 2550500
3343  Utot 2672866
3344  Utot 2589414
3345  Utot 2734106
3346  Utot 2609759
3347  Utot 2576227
3348  Utot 2859945
3349  Utot 2780620
3350  Utot 2893274
3351  Utot 2554782
3352  Utot 2827353
3353  Utot 2770437
3354  Utot 2562350
3355  Utot 2798051
3356  Utot 2800162
3357  Utot 2660401
3358  Utot 2769410
3359  Utot 2830389
3360  Utot 2891782
3361  Utot 2792942
3362  Utot 2632520
3363  Utot 2734529
3364  Utot 2802269
3365  Utot 2661187
3366  Utot 2535155
3367  Utot 2664198
3368  Utot 2711589
3369  Utot 2685195
3370  Utot 2789444
3371  Utot 2583074
3372  Utot 2647040
3373  Utot 2851269
3374  Utot 2704929
3375  Utot 2758286
3376  Utot 2727875
3377  Utot 2656128
3378  Utot 2675579
3379  Utot 2690179
3380  Utot 2608122
3381  Utot 2574141
3382  Utot 2569043
3383  Utot 2640780
3384  Utot 2658874
3385  Utot 2687802
3386  Utot 2871389
3387  Utot 2582433
3388  Utot 2689726
3389  Utot 2627489
3390  Utot 2699148
3391  Utot 2847554
3392  Utot 2583111
3393  Utot 2716563
3394  Utot 2794584
3395  Utot 2629455
3396  Utot 2722197
3397  Utot 2681719
3398  Utot 2566169
3399  Utot 2762546
3400  Utot 2694113
3401  Utot 2774003
3402  Utot 2608083
3403  Utot 2714489
3404  Utot 2871973
3405  Utot 2798553
3406  Utot 2757534
3407  Utot 2798637
3408  Utot 2765725
3409  Utot 2522749
3410  Utot 2694330
3411  Utot 2690594
3412  Utot 2645348
3413  Utot 2727054
3414  Utot 2758074
3415  Utot 2745063
3416  Utot 2746636
3417  Utot 2856461
3418  Utot 2684350
3419  Utot 2560002
3420  Utot 2619008
3421  Utot 2637049
3422  Utot 2833207
3423  Utot 2774882
3424  Utot 2980105
3425  Utot 2897293
3426  Utot 2629372
3427  Utot 2592530
3428  Utot 2693271
3429  Utot 2508924
3430  Utot 2648043
3431  Utot 2906156
3432  Utot 2591477
3433  Utot 2519544
3434  Utot 2870429
3435  Utot 2825686
3436  Utot 2753483
3437  Utot 2517856
3438  Utot 2748610
3439  Utot 2639650
3440  Utot 2781677
3441  Utot 2702265
3442  Utot 2680730
3443  Utot 2580857
3444  Utot 2561568
3445  Utot 2795381
3446  Utot 2627217
3447  Utot 2652016
3448  Utot 2563328
3449  Utot 2754500
3450  Utot 2842736
3451  Utot 2866005
3452  Utot 2761392
3453  Utot 2662856
3454  Utot 2818088
3455  Utot 2594907
3456  Utot 2883328
3457  Utot 2716775
3458  Utot 2697577
3459  Utot 2689067
3460  Utot 2673328
3461  Utot 2668918
3462  Utot 2722666
3463  Utot 2711179
3464  Utot 2621000
3465  Utot 2669950
3466  Utot 2638697
3467  Utot 2713771
3468  Utot 2697229
3469  Utot 2758472
3470  Utot 2903225
3471  Utot 2660320
3472  Utot 2703437
3473  Utot 2595270
3474  Utot 2558686
3475  Utot 2779014
3476  Utot 2826439
3477  Utot 2537512
3478  Utot 2602842
3479  Utot 2705165
3480  Utot 2528797
3481  Utot 2804126
3482  Utot 2857781
3483  Utot 2756156
3484  Utot 2788715
3485  Utot 2657847
3486  Utot 2742490
3487  Utot 2741672
3488  Utot 2629992
3489  Utot 2745165
3490  Utot 2674684
3491  Utot 3024558
3492  Utot 2822362
3493  Utot 2815161
3494  Utot 2664001
3495  Utot 2661271
3496  Utot 2723889
3497  Utot 2522107
3498  Utot 2935821
3499  Utot 2719761
3500  Utot 2861760
3501  Utot 2793897
3502  Utot 2844261
3503  Utot 2650811
3504  Utot 2653463
3505  Utot 2728434
3506  Utot 2745208
3507  Utot 2660599
3508  Utot 2768965
3509  Utot 2662025
3510  Utot 2795574
3511  Utot 2729579
3512  Utot 2751717
3513  Utot 2664084
3514  Utot 2687019
3515  Utot 2885015
3516  Utot 2646594
3517  Utot 2687394
3518  Utot 2491994
3519  Utot 2714473
3520  Utot 2685186
3521  Utot 2728812
3522  Utot 2600076
3523  Utot 2651640
3524  Utot 2687203
3525  Utot 2558586
3526  Utot 2680884
3527  Utot 2668967
3528  Utot 2618662
3529  Utot 2765641
3530  Utot 2707468
3531  Utot 2656272
3532  Utot 2658685
3533  Utot 2808869
3534  Utot 2722614
3535  Utot 2625964
3536  Utot 2710792
3537  Utot 2646567
3538  Utot 2832407
3539  Utot 2740231
3540  Utot 2719199
3541  Utot 2833520
3542  Utot 2575965
3543  Utot 2690728
3544  Utot 2574625
3545  Utot 2804026
3546  Utot 2843524
3547  Utot 2564670
3548  Utot 2879340
3549  Utot 2680683
3550  Utot 2675638
3551  Utot 2972124
3552  Utot 2715324
3553  Utot 2718451
3554  Utot 2687537
3555  Utot 2824345
3556  Utot 2658592
3557  Utot 2833847
3558  Utot 2692415
3559  Utot 2694078
3560  Utot 2509261
3561  Utot 2844971
3562  Utot 2985524
3563  Utot 2734523
3564  Utot 2568009
3565  Utot 2703651
3566  Utot 2736591
3567  Utot 2728049
3568  Utot 2564850
3569  Utot 2804949
3570  Utot 2666287
3571  Utot 2610776
3572  Utot 2765853
3573  Utot 2698928
3574  Utot 2669502
3575  Utot 2742229
3576  Utot 2657974
3577  Utot 2574381
3578  Utot 2861913
3579  Utot 2657729
3580  Utot 2749740
3581  Utot 2654356
3582  Utot 2850983
3583  Utot 2560086
3584  Utot 2885416
3585  Utot 2718142
3586  Utot 2577808
3587  Utot 2713786
3588  Utot 2670356
3589  Utot 2814699
3590  Utot 2792041
3591  Utot 2817743
3592  Utot 2745804
3593  Utot 2699853
3594  Utot 2686188
3595  Utot 2515356
3596  Utot 2591159
3597  Utot 2670475
3598  Utot 2784239
3599  Utot 2581617
3600  Utot 2685219
3601  Utot 2705796
3602  Utot 2714392
3603  Utot 2752790
3604  Utot 2621762
3605  Utot 2657676
3606  Utot 2681644
3607  Utot 2926236
3608  Utot 2777324
3609  Utot 2819542
3610  Utot 2672324
3611  Utot 2681155
3612  Utot 2952723
3613  Utot 2753294
3614  Utot 2747255
3615  Utot 2868510
3616  Utot 2539338
3617  Utot 2697930
3618  Utot 2688269
3619  Utot 2556621
3620  Utot 2724666
3621  Utot 2595298
3622  Utot 2790962
3623  Utot 2698642
3624  Utot 2885009
3625  Utot 2664581
3626  Utot 2810524
3627  Utot 2688647
3628  Utot 2589355
3629  Utot 2697823
3630  Utot 2931154
3631  Utot 2596380
3632  Utot 2743303
3633  Utot 2549874
3634  Utot 2598797
3635  Utot 2773082
3636  Utot 2655436
3637  Utot 2718947
3638  Utot 2573321
3639  Utot 2840013
3640  Utot 2716901
3641  Utot 2794913
3642  Utot 2464026
3643  Utot 2621654
3644  Utot 2863368
3645  Utot 2905623
3646  Utot 2729576
3647  Utot 2670959
3648  Utot 2770366
3649  Utot 2751725
3650  Utot 2638835
3651  Utot 2616592
3652  Utot 2733286
3653  Utot 2648630
3654  Utot 2785422
3655  Utot 2824497
3656  Utot 2652891
3657  Utot 2704147
3658  Utot 2580123
3659  Utot 2574434
3660  Utot 2691683
3661  Utot 2610805
3662  Utot 2742790
3663  Utot 2770470
3664  Utot 2565739
3665  Utot 2646353
3666  Utot 2792291
3667  Utot 2712908
3668  Utot 2679693
3669  Utot 2562469
3670  Utot 2707845
3671  Utot 2687178
3672  Utot 2532924
3673  Utot 2610535
3674  Utot 2640048
3675  Utot 2570698
3676  Utot 2667359
3677  Utot 2612430
3678  Utot 2662416
3679  Utot 2802605
3680  Utot 2716184
3681  Utot 2686894
3682  Utot 2850023
3683  Utot 2673437
3684  Utot 2797641
3685  Utot 2804619
3686  Utot 2693745
3687  Utot 2681100
3688  Utot 2694997
3689  Utot 2707515
3690  Utot 2538583
3691  Utot 2867910
3692  Utot 2659528
3693  Utot 2704768
3694  Utot 2773949
3695  Utot 2800872
3696  Utot 2460992
3697  Utot 2692843
3698  Utot 2739944
3699  Utot 2714533
3700  Utot 2641285
3701  Utot 2622031
3702  Utot 2811311
3703  Utot 2983024
3704  Utot 2679425
3705  Utot 2934651
3706  Utot 2898703
3707  Utot 2712583
3708  Utot 2681973
3709  Utot 2735186
3710  Utot 2717923
3711  Utot 2787279
3712  Utot 2841537
3713  Utot 2655694
3714  Utot 2957548
3715  Utot 2619120
3716  Utot 2786861
3717  Utot 2667654
3718  Utot 2514984
3719  Utot 2655761
3720  Utot 2762995
3721  Utot 2905679
3722  Utot 2780953
3723  Utot 2776398
3724  Utot 2786533
3725  Utot 2585720
3726  Utot 2610907
3727  Utot 2769260
3728  Utot 2790485
3729  Utot 2703742
3730  Utot 2774132
3731  Utot 2624839
3732  Utot 2946200
3733  Utot 2714026
3734  Utot 2821812
3735  Utot 2516140
3736  Utot 2864472
3737  Utot 2756918
3738  Utot 2614712
3739  Utot 2641690
3740  Utot 2604271
3741  Utot 2687839
3742  Utot 2687669
3743  Utot 2606426
3744  Utot 2556790
3745  Utot 2672603
3746  Utot 2639592
3747  Utot 2913240
3748  Utot 2523182
3749  Utot 2638732
3750  Utot 2577304
3751  Utot 2750776
3752  Utot 2710152
3753  Utot 2743881
3754  Utot 2586857
3755  Utot 2685205
3756  Utot 2712875
3757  Utot 2716011
3758  Utot 2716648
3759  Utot 2709002
3760  Utot 2747864
3761  Utot 2481524
3762  Utot 2753844
3763  Utot 2630427
3764  Utot 2568117
3765  Utot 2714825
3766  Utot 2778888
3767  Utot 2538625
3768  Utot 2874860
3769  Utot 2702445
3770  Utot 2711289
3771  Utot 2637297
3772  Utot 2926879
3773  Utot 2687236
3774  Utot 2681277
3775  Utot 2929240
3776  Utot 2683371
3777  Utot 2719200
3778  Utot 2758654
3779  Utot 2723955
3780  Utot 2605435
3781  Utot 2646852
3782  Utot 2611821
3783  Utot 2619661
3784  Utot 2748776
3785  Utot 2816123
3786  Utot 2781350
3787  Utot 2771920
3788  Utot 2623450
3789  Utot 2616670
3790  Utot 2817148
3791  Utot 2756076
3792  Utot 2780033
3793  Utot 2588621
3794  Utot 2686620
3795  Utot 2721602
3796  Utot 2706384
3797  Utot 2721314
3798  Utot 2828731
3799  Utot 2659431
3800  Utot 2708327
3801  Utot 2668482
3802  Utot 2678707
3803  Utot 2584633
3804  Utot 3005231
3805  Utot 2644194
3806  Utot 2672674
3807  Utot 2721638
3808  Utot 2805305
3809  Utot 2892118
3810  Utot 2654994
3811  Utot 2734668
3812  Utot 2674601
3813  Utot 2763881
3814  Utot 2599829
3815  Utot 2578334
3816  Utot 2529336
3817  Utot 2690313
3818  Utot 2749187
3819  Utot 2653187
3820  Utot 2561580
3821  Utot 2729427
3822  Utot 2818355
3823  Utot 2793147
3824  Utot 2618666
3825  Utot 2746918
3826  Utot 2723405
3827  Utot 2529261
3828  Utot 2819002
3829  Utot 2721147
3830  Utot 2594118
3831  Utot 2739792
3832  Utot 2840551
3833  Utot 2810780
3834  Utot 2699813
3835  Utot 2741648
3836  Utot 2879651
3837  Utot 2721537
3838  Utot 2778123
3839  Utot 2754787
3840  Utot 2593171
3841  Utot 2623078
3842  Utot 2646432
3843  Utot 2763805
3844  Utot 2710820
3845  Utot 2695580
3846  Utot 2819179
3847  Utot 2651254
3848  Utot 2876605
3849  Utot 2572575
3850  Utot 2751333
3851  Utot 2580913
3852  Utot 2657689
3853  Utot 2662872
3854  Utot 2727596
3855  Utot 2878858
3856  Utot 2800577
3857  Utot 2860738
3858  Utot 2675010
3859  Utot 2627258
3860  Utot 2632325
3861  Utot 2632859
3862  Utot 2591805
3863  Utot 3007042
3864  Utot 2751924
3865  Utot 2593610
3866  Utot 2809678
3867  Utot 2643269
3868  Utot 2763653
3869  Utot 2719812
3870  Utot 2746372
3871  Utot 2699876
3872  Utot 2757870
3873  Utot 2745932
3874  Utot 2754746
3875  Utot 2785609
3876  Utot 2640312
3877  Utot 2732304
3878  Utot 2661937
3879  Utot 2542490
3880  Utot 2661902
3881  Utot 2661860
3882  Utot 2581329
3883  Utot 2741929
3884  Utot 2570898
3885  Utot 2579573
3886  Utot 2944383
3887  Utot 2796659
3888  Utot 2788765
3889  Utot 2618938
3890  Utot 2626571
3891  Utot 2559972
3892  Utot 2626742
3893  Utot 2750509
3894  Utot 2809646
3895  Utot 2931939
3896  Utot 2702406
3897  Utot 2629968
3898  Utot 2612693
3899  Utot 2670492
3900  Utot 2589128
3901  Utot 2652454
3902  Utot 2746687
3903  Utot 2516620
3904  Utot 2509805
3905  Utot 2772317
3906  Utot 2771110
3907  Utot 2623542
3908  Utot 2664750
3909  Utot 2837487
3910  Utot 2798337
3911  Utot 2754753
3912  Utot 2680695
3913  Utot 2740246
3914  Utot 2659028
3915  Utot 2716198
3916  Utot 2866740
3917  Utot 2767648
3918  Utot 2994117
3919  Utot 2734737
3920  Utot 2725405
3921  Utot 2637113
3922  Utot 2631906
3923  Utot 2647752
3924  Utot 2685440
3925  Utot 2820395
3926  Utot 2632496
3927  Utot 2897366
3928  Utot 2747188
3929  Utot 2789892
3930  Utot 2831883
3931  Utot 2540058
3932  Utot 2789655
3933  Utot 2715182
3934  Utot 2667775
3935  Utot 2754004
3936  Utot 2833461
3937  Utot 2875152
3938  Utot 2768416
3939  Utot 2626842
3940  Utot 2738667
3941  Utot 2720235
3942  Utot 2669507
3943  Utot 2574234
3944  Utot 2625225
3945  Utot 2769059
3946  Utot 2576873
3947  Utot 2745430
3948  Utot 2965732
3949  Utot 2634203
3950  Utot 2838141
3951  Utot 2804159
3952  Utot 2785506
3953  Utot 2897542
3954  Utot 2750490
3955  Utot 2772239
3956  Utot 2746792
3957  Utot 2764776
3958  Utot 2703023
3959  Utot 2518342
3960  Utot 2883507
3961  Utot 2630726
3962  Utot 2877413
3963  Utot 2744835
3964  Utot 2802642
3965  Utot 2630052
3966  Utot 2743319
3967  Utot 2670463
3968  Utot 2879161
3969  Utot 2590677
3970  Utot 2639307
3971  Utot 2788224
3972  Utot 2794000
3973  Utot 2704849
3974  Utot 2623970
3975  Utot 2695663
3976  Utot 2920341
3977  Utot 2630806
3978  Utot 2458344
3979  Utot 2698603
3980  Utot 2706065
3981  Utot 2736629
3982  Utot 2677345
3983  Utot 2708087
3984  Utot 2811897
3985  Utot 2653362
3986  Utot 2858453
3987  Utot 2654476
3988  Utot 2694729
3989  Utot 2593536
3990  Utot 2658914
3991  Utot 2575731
3992  Utot 2623955
3993  Utot 2663471
3994  Utot 2838967
3995  Utot 2648586
3996  Utot 2782953
3997  Utot 2647932
3998  Utot 2591944
3999  Utot 2781419
4000  Utot 2811001
4001  Utot 2741028
4002  Utot 2893322
4003  Utot 2736575
4004  Utot 2716916
4005  Utot 2561813
4006  Utot 2711812
4007  Utot 2777610
4008  Utot 2493939
4009  Utot 2712732
4010  Utot 2772044
4011  Utot 2797473
4012  Utot 2732737
4013  Utot 2811992
4014  Utot 2695016
4015  Utot 2743701
4016  Utot 2611606
4017  Utot 2816098
4018  Utot 2650335
4019  Utot 2749333
4020  Utot 3010652
4021  Utot 2565416
4022  Utot 2817646
4023  Utot 2507207
4024  Utot 2542766
4025  Utot 2923604
4026  Utot 2754943
4027  Utot 2682966
4028  Utot 2659029
4029  Utot 2762314
4030  Utot 2721115
4031  Utot 2850100
4032  Utot 2689256
4033  Utot 2746526
4034  Utot 2612776
4035  Utot 2444836
4036  Utot 2662762
4037  Utot 2641174
4038  Utot 2802887
4039  Utot 2679641
4040  Utot 2679864
4041  Utot 2842520
4042  Utot 2803700
4043  Utot 2543157
4044  Utot 2746670
4045  Utot 2695776
4046  Utot 2767922
4047  Utot 2643677
4048  Utot 2749348
4049  Utot 2749950
4050  Utot 2756025
4051  Utot 2581567
4052  Utot 2749720
4053  Utot 2687319
4054  Utot 2666861
4055  Utot 2738183
4056  Utot 2702225
4057  Utot 2637285
4058  Utot 2777172
4059  Utot 2641786
4060  Utot 2801116
4061  Utot 2712741
4062  Utot 2617697
4063  Utot 2451816
4064  Utot 2810807
4065  Utot 2698664
4066  Utot 2749770
4067  Utot 2954199
4068  Utot 2652058
4069  Utot 2652334
4070  Utot 2788357
4071  Utot 2815958
4072  Utot 2599901
4073  Utot 2646509
4074  Utot 2601057
4075  Utot 2939711
4076  Utot 2648724
4077  Utot 2810867
4078  Utot 2772574
4079  Utot 2773357
4080  Utot 2612342
4081  Utot 2722701
4082  Utot 2722755
4083  Utot 2839408
4084  Utot 2736709
4085  Utot 2636908
4086  Utot 2806499
4087  Utot 2584179
4088  Utot 2776549
4089  Utot 2720490
4090  Utot 2760046
4091  Utot 2812517
4092  Utot 2826609
4093  Utot 2822984
4094  Utot 2591521
4095  Utot 2566930
4096  Utot 2762678
4097  Utot 2631161
4098  Utot 2622654
4099  Utot 2591430
4100  Utot 2666929
4101  Utot 2746391
4102  Utot 2710928
4103  Utot 2684068
4104  Utot 2644047
4105  Utot 2419792
4106  Utot 2694959
4107  Utot 2809336
4108  Utot 2793578
4109  Utot 2779465
4110  Utot 2750530
4111  Utot 2991762
4112  Utot 2697213
4113  Utot 2730758
4114  Utot 2423462
4115  Utot 2612781
4116  Utot 2831488
4117  Utot 2579166
4118  Utot 2797761
4119  Utot 2788431
4120  Utot 2946709
4121  Utot 2769367
4122  Utot 2692824
4123  Utot 2610389
4124  Utot 2601489
4125  Utot 2671039
4126  Utot 2833971
4127  Utot 3031499
4128  Utot 2641847
4129  Utot 2724162
4130  Utot 2817529
4131  Utot 2704229
4132  Utot 2580075
4133  Utot 2778392
4134  Utot 2753012
4135  Utot 2784953
4136  Utot 2793292
4137  Utot 2702468
4138  Utot 2738162
4139  Utot 2715646
4140  Utot 2487752
4141  Utot 2684660
4142  Utot 2730819
4143  Utot 2584883
4144  Utot 2682260
4145  Utot 2703215
4146  Utot 2474606
4147  Utot 2774409
4148  Utot 2658929
4149  Utot 2599252
4150  Utot 2443754
4151  Utot 2694422
4152  Utot 2677742
4153  Utot 2649999
4154  Utot 2583655
4155  Utot 2792926
4156  Utot 2752898
4157  Utot 2799944
4158  Utot 2672335
4159  Utot 2978234
4160  Utot 2844256
4161  Utot 2704426
4162  Utot 2667598
4163  Utot 2653646
4164  Utot 2561872
4165  Utot 2665327
4166  Utot 2514080
4167  Utot 2721306
4168  Utot 2906557
4169  Utot 2607518
4170  Utot 2754210
4171  Utot 2808152
4172  Utot 2777134
4173  Utot 2730773
4174  Utot 2819852
4175  Utot 2583125
4176  Utot 2663066
4177  Utot 2671563
4178  Utot 2789529
4179  Utot 2826505
4180  Utot 2624220
4181  Utot 2781327
4182  Utot 2723138
4183  Utot 2554272
4184  Utot 2730696
4185  Utot 2667409
4186  Utot 2696887
4187  Utot 2766015
4188  Utot 2717435
4189  Utot 2749464
4190  Utot 2767288
4191  Utot 2656613
4192  Utot 2708495
4193  Utot 2646293
4194  Utot 2762839
4195  Utot 2603897
4196  Utot 2736841
4197  Utot 2681486
4198  Utot 2673688
4199  Utot 2785105
4200  Utot 2655605
4201  Utot 2519923
4202  Utot 2882913
4203  Utot 2476639
4204  Utot 2709088
4205  Utot 2567997
4206  Utot 2751401
4207  Utot 2651074
4208  Utot 2652394
4209  Utot 2563900
4210  Utot 2774235
4211  Utot 2576153
4212  Utot 2818669
4213  Utot 2621351
4214  Utot 2624345
4215  Utot 2549995
4216  Utot 2584691
4217  Utot 2833159
4218  Utot 2606320
4219  Utot 2810390
4220  Utot 2666585
4221  Utot 2713440
4222  Utot 2690090
4223  Utot 2694737
4224  Utot 2747932
4225  Utot 2803847
4226  Utot 2685388
4227  Utot 2713390
4228  Utot 2703337
4229  Utot 2776466
4230  Utot 2592909
4231  Utot 2754688
4232  Utot 2707771
4233  Utot 2763755
4234  Utot 2867755
4235  Utot 2690005
4236  Utot 2668556
4237  Utot 2703981
4238  Utot 2744913
4239  Utot 2539538
4240  Utot 2709233
4241  Utot 2824492
4242  Utot 2772966
4243  Utot 2557598
4244  Utot 2637602
4245  Utot 2843126
4246  Utot 2870534
4247  Utot 2747128
4248  Utot 2770294
4249  Utot 2779278
4250  Utot 2697633
4251  Utot 2706401
4252  Utot 2812775
4253  Utot 2721127
4254  Utot 2695697
4255  Utot 2754245
4256  Utot 2809531
4257  Utot 2695758
4258  Utot 2755016
4259  Utot 2681027
4260  Utot 2723224
4261  Utot 2651321
4262  Utot 2719822
4263  Utot 2661116
4264  Utot 2660919
4265  Utot 2538398
4266  Utot 2715300
4267  Utot 2646945
4268  Utot 2729089
4269  Utot 2755265
4270  Utot 2421144
4271  Utot 2670876
4272  Utot 2984868
4273  Utot 2754719
4274  Utot 3061306
4275  Utot 2774983
4276  Utot 2632797
4277  Utot 2820604
4278  Utot 2585950
4279  Utot 2615942
4280  Utot 2731869
4281  Utot 2697770
4282  Utot 2803817
4283  Utot 2888849
4284  Utot 2884739
4285  Utot 2593602
4286  Utot 2735343
4287  Utot 2745548
4288  Utot 2789216
4289  Utot 2840215
4290  Utot 2629396
4291  Utot 2797141
4292  Utot 2734943
4293  Utot 2644977
4294  Utot 2580593
4295  Utot 2792530
4296  Utot 2715074
4297  Utot 2691412
4298  Utot 2534253
4299  Utot 2617709
4300  Utot 2810200
4301  Utot 2890225
4302  Utot 2611430
4303  Utot 2742524
4304  Utot 2691748
4305  Utot 2617690
4306  Utot 2751044
4307  Utot 2649979
4308  Utot 2779077
4309  Utot 2704379
4310  Utot 2966401
4311  Utot 2733250
4312  Utot 2823421
4313  Utot 2797586
4314  Utot 2772268
4315  Utot 2638052
4316  Utot 2603466
4317  Utot 2623679
4318  Utot 2782283
4319  Utot 2701180
4320  Utot 2723726
4321  Utot 2753959
4322  Utot 2694054
4323  Utot 2600965
4324  Utot 2905091
4325  Utot 2685023
4326  Utot 2734247
4327  Utot 2807912
4328  Utot 2842878
4329  Utot 2928306
4330  Utot 2673861
4331  Utot 2738156
4332  Utot 2831379
4333  Utot 2714049
4334  Utot 2772243
4335  Utot 2690129
4336  Utot 2650273
4337  Utot 2608904
4338  Utot 2567016
4339  Utot 2857217
4340  Utot 2708466
4341  Utot 2795032
4342  Utot 2787212
4343  Utot 2762202
4344  Utot 2670093
4345  Utot 2635525
4346  Utot 2763653
4347  Utot 2792369
4348  Utot 2602495
4349  Utot 2661143
4350  Utot 2734376
4351  Utot 2590498
4352  Utot 2770343
4353  Utot 2746083
4354  Utot 2545835
4355  Utot 2625261
4356  Utot 2750683
4357  Utot 2719509
4358  Utot 2747435
4359  Utot 2856961
4360  Utot 2598334
4361  Utot 2600676
4362  Utot 2645720
4363  Utot 2831293
4364  Utot 2747212
4365  Utot 2597862
4366  Utot 2759310
4367  Utot 2620497
4368  Utot 2679729
4369  Utot 2645662
4370  Utot 2656419
4371  Utot 2589147
4372  Utot 2735480
4373  Utot 2844684
4374  Utot 2770937
4375  Utot 2743326
4376  Utot 2795631
4377  Utot 2665532
4378  Utot 2778380
4379  Utot 2631639
4380  Utot 2835472
4381  Utot 2862734
4382  Utot 2809433
4383  Utot 2819873
4384  Utot 2752381
4385  Utot 2860011
4386  Utot 2973671
4387  Utot 2712083
4388  Utot 2764391
4389  Utot 2652999
4390  Utot 2608606
4391  Utot 2641160
4392  Utot 2729447
4393  Utot 2543405
4394  Utot 2703997
4395  Utot 2763326
4396  Utot 2797583
4397  Utot 2758281
4398  Utot 2771003
4399  Utot 2692081
4400  Utot 2776343
4401  Utot 2792986
4402  Utot 2666316
4403  Utot 2643858
4404  Utot 2592436
4405  Utot 2774516
4406  Utot 2637740
4407  Utot 2654223
4408  Utot 2682110
4409  Utot 2560294
4410  Utot 2695801
4411  Utot 2728731
4412  Utot 2555916
4413  Utot 2609674
4414  Utot 2785037
4415  Utot 2582855
4416  Utot 2884710
4417  Utot 2720477
4418  Utot 2538617
4419  Utot 2550653
4420  Utot 2773766
4421  Utot 2708826
4422  Utot 2625644
4423  Utot 2504646
4424  Utot 2680987
4425  Utot 2345681
4426  Utot 2649871
4427  Utot 2752046
4428  Utot 2621295
4429  Utot 2610765
4430  Utot 2635398
4431  Utot 2718854
4432  Utot 2708791
4433  Utot 2707360
4434  Utot 2664080
4435  Utot 2686626
4436  Utot 2775178
4437  Utot 2708700
4438  Utot 2791352
4439  Utot 2884399
4440  Utot 2512537
4441  Utot 2697822
4442  Utot 2735376
4443  Utot 2591566
4444  Utot 2663009
4445  Utot 2668523
4446  Utot 2565491
4447  Utot 2935251
4448  Utot 2603261
4449  Utot 2587987
4450  Utot 2569461
4451  Utot 2603288
4452  Utot 2839057
4453  Utot 2830353
4454  Utot 2642956
4455  Utot 2649408
4456  Utot 2653187
4457  Utot 2773424
4458  Utot 2749870
4459  Utot 2792155
4460  Utot 2708147
4461  Utot 2601312
4462  Utot 2532881
4463  Utot 2857916
4464  Utot 2748427
4465  Utot 2548706
4466  Utot 2934147
4467  Utot 2737105
4468  Utot 2695684
4469  Utot 2872914
4470  Utot 2690658
4471  Utot 2884455
4472  Utot 2841880
4473  Utot 2646289
4474  Utot 2698878
4475  Utot 2674264
4476  Utot 2626076
4477  Utot 2820483
4478  Utot 2576064
4479  Utot 2840880
4480  Utot 2643438
4481  Utot 2660264
4482  Utot 2880729
4483  Utot 3009382
4484  Utot 2815830
4485  Utot 2828111
4486  Utot 2703845
4487  Utot 2720080
4488  Utot 2725358
4489  Utot 2909556
4490  Utot 2668366
4491  Utot 2619610
4492  Utot 2645735
4493  Utot 2569775
4494  Utot 2547404
4495  Utot 2697886
4496  Utot 2742378
4497  Utot 2769009
4498  Utot 2472219
4499  Utot 2792021
4500  Utot 2747682
4501  Utot 2648034
4502  Utot 2661090
4503  Utot 2871402
4504  Utot 2673347
4505  Utot 2671217
4506  Utot 2788865
4507  Utot 2720231
4508  Utot 2671843
4509  Utot 2737722
4510  Utot 2671539
4511  Utot 2693889
4512  Utot 2690434
4513  Utot 2583818
4514  Utot 2551096
4515  Utot 2699757
4516  Utot 2813678
4517  Utot 2836744
4518  Utot 2736904
4519  Utot 2657879
4520  Utot 2804596
4521  Utot 2736526
4522  Utot 2689014
4523  Utot 2974481
4524  Utot 2819371
4525  Utot 2622729
4526  Utot 2774938
4527  Utot 2703311
4528  Utot 2606072
4529  Utot 2734795
4530  Utot 2719278
4531  Utot 2777052
4532  Utot 2836879
4533  Utot 2693790
4534  Utot 2744249
4535  Utot 2735606
4536  Utot 2914782
4537  Utot 2588155
4538  Utot 2714598
4539  Utot 2863114
4540  Utot 2661355
4541  Utot 2593750
4542  Utot 2907745
4543  Utot 2726969
4544  Utot 2673000
4545  Utot 2665259
4546  Utot 2722260
4547  Utot 2644625
4548  Utot 2720716
4549  Utot 2752324
4550  Utot 2705350
4551  Utot 2721456
4552  Utot 2914295
4553  Utot 2723847
4554  Utot 2750578
4555  Utot 2803869
4556  Utot 2734256
4557  Utot 2621408
4558  Utot 2754285
4559  Utot 2869582
4560  Utot 2559298
4561  Utot 2662190
4562  Utot 2597820
4563  Utot 2635891
4564  Utot 2698338
4565  Utot 2864510
4566  Utot 2819323
4567  Utot 2674801
4568  Utot 2727368
4569  Utot 2766565
4570  Utot 2746466
4571  Utot 2669211
4572  Utot 2895777
4573  Utot 2524986
4574  Utot 2780946
4575  Utot 2534981
4576  Utot 2786139
4577  Utot 2696923
4578  Utot 2801760
4579  Utot 2708592
4580  Utot 2751103
4581  Utot 2619548
4582  Utot 2721589
4583  Utot 2499486
4584  Utot 2532648
4585  Utot 2747687
4586  Utot 2791997
4587  Utot 2727224
4588  Utot 2851725
4589  Utot 2629481
4590  Utot 2579661
4591  Utot 2762490
4592  Utot 2695079
4593  Utot 2792703
4594  Utot 2804245
4595  Utot 2778312
4596  Utot 2535983
4597  Utot 2924855
4598  Utot 2812227
4599  Utot 2738333
4600  Utot 2689342
4601  Utot 2665958
4602  Utot 2705600
4603  Utot 2695475
4604  Utot 2617969
4605  Utot 2795973
4606  Utot 2675347
4607  Utot 2652899
4608  Utot 2815193
4609  Utot 2740822
4610  Utot 2692277
4611  Utot 2653863
4612  Utot 2807634
4613  Utot 2679907
4614  Utot 2759049
4615  Utot 2640021
4616  Utot 2745529
4617  Utot 2685568
4618  Utot 2600747
4619  Utot 2883971
4620  Utot 2755667
4621  Utot 2740567
4622  Utot 2604841
4623  Utot 2593651
4624  Utot 2669536
4625  Utot 2530971
4626  Utot 2913227
4627  Utot 2733624
4628  Utot 2609410
4629  Utot 2544683
4630  Utot 2621922
4631  Utot 2809624
4632  Utot 2675435
4633  Utot 2675570
4634  Utot 2536041
4635  Utot 2681266
4636  Utot 2696655
4637  Utot 2718861
4638  Utot 2758393
4639  Utot 2801009
4640  Utot 2564230
4641  Utot 3030380
4642  Utot 2736422
4643  Utot 2802694
4644  Utot 2575816
4645  Utot 2769501
4646  Utot 2631187
4647  Utot 2757092
4648  Utot 2613080
4649  Utot 2527134
4650  Utot 2670055
4651  Utot 2595905
4652  Utot 2607640
4653  Utot 2840077
4654  Utot 2892176
4655  Utot 2764805
4656  Utot 2666556
4657  Utot 2821653
4658  Utot 2746566
4659  Utot 2730701
4660  Utot 2775366
4661  Utot 2832423
4662  Utot 2790203
4663  Utot 2767420
4664  Utot 2743091
4665  Utot 2656296
4666  Utot 2827562
4667  Utot 2749379
4668  Utot 2701764
4669  Utot 2516980
4670  Utot 3001073
4671  Utot 2687542
4672  Utot 2787056
4673  Utot 2607839
4674  Utot 2702036
4675  Utot 2636616
4676  Utot 2528263
4677  Utot 2661303
4678  Utot 2611529
4679  Utot 2627912
4680  Utot 2844224
4681  Utot 2585000
4682  Utot 2686448
4683  Utot 2706677
4684  Utot 2694255
4685  Utot 2782805
4686  Utot 2579421
4687  Utot 2658754
4688  Utot 2715906
4689  Utot 2667967
4690  Utot 2761517
4691  Utot 2692533
4692  Utot 2730956
4693  Utot 2491699
4694  Utot 2791405
4695  Utot 2716434
4696  Utot 2686605
4697  Utot 2762408
4698  Utot 2916344
4699  Utot 2493396
4700  Utot 2666074
4701  Utot 2607974
4702  Utot 2660550
4703  Utot 2536378
4704  Utot 2769810
4705  Utot 2729290
4706  Utot 2673932
4707  Utot 2518666
4708  Utot 2746193
4709  Utot 2628774
4710  Utot 2656567
4711  Utot 2754521
4712  Utot 2724109
4713  Utot 2811094
4714  Utot 2764112
4715  Utot 2660351
4716  Utot 2728928
4717  Utot 2542085
4718  Utot 2527458
4719  Utot 2724380
4720  Utot 2804934
4721  Utot 2925205
4722  Utot 2777964
4723  Utot 2548404
4724  Utot 2867130
4725  Utot 2850501
4726  Utot 2886920
4727  Utot 2619910
4728  Utot 2706155
4729  Utot 2661379
4730  Utot 2629449
4731  Utot 2731984
4732  Utot 2752771
4733  Utot 2485568
4734  Utot 2691770
4735  Utot 2583618
4736  Utot 2526370
4737  Utot 2589631
4738  Utot 2699403
4739  Utot 2701242
4740  Utot 2758186
4741  Utot 2695163
4742  Utot 2726210
4743  Utot 2710162
4744  Utot 2456184
4745  Utot 2715378
4746  Utot 2771699
4747  Utot 2773632
4748  Utot 2728429
4749  Utot 2734065
4750  Utot 2732570
4751  Utot 2628245
4752  Utot 2737860
4753  Utot 2695277
4754  Utot 2556757
4755  Utot 2778698
4756  Utot 2831627
4757  Utot 2776479
4758  Utot 2801668
4759  Utot 2783479
4760  Utot 2832202
4761  Utot 2704347
4762  Utot 2763793
4763  Utot 2973497
4764  Utot 2731434
4765  Utot 2600865
4766  Utot 2690099
4767  Utot 2564825
4768  Utot 2767770
4769  Utot 2729063
4770  Utot 2424559
4771  Utot 2689653
4772  Utot 2605820
4773  Utot 2545800
4774  Utot 2921458
4775  Utot 2653182
4776  Utot 2876124
4777  Utot 2773698
4778  Utot 2823463
4779  Utot 2610736
4780  Utot 2662885
4781  Utot 2729972
4782  Utot 2816055
4783  Utot 2684616
4784  Utot 2560164
4785  Utot 2699183
4786  Utot 2718704
4787  Utot 2740209
4788  Utot 2677576
4789  Utot 2831414
4790  Utot 2567214
4791  Utot 2751354
4792  Utot 2833245
4793  Utot 2814619
4794  Utot 2763791
4795  Utot 2752218
4796  Utot 2800022
4797  Utot 2575651
4798  Utot 2633435
4799  Utot 2608761
4800  Utot 2808106
4801  Utot 2867441
4802  Utot 2648925
4803  Utot 2915273
4804  Utot 2630207
4805  Utot 2632688
4806  Utot 2725294
4807  Utot 2696124
4808  Utot 2770155
4809  Utot 2726356
4810  Utot 2830336
4811  Utot 2817388
4812  Utot 2553154
4813  Utot 2567098
4814  Utot 2738302
4815  Utot 2677370
4816  Utot 2771800
4817  Utot 2626528
4818  Utot 2656410
4819  Utot 2551712
4820  Utot 2633735
4821  Utot 2641765
4822  Utot 2618053
4823  Utot 2631537
4824  Utot 2808993
4825  Utot 2804457
4826  Utot 2792389
4827  Utot 2773514
4828  Utot 2747393
4829  Utot 2658887
4830  Utot 2767569
4831  Utot 2778683
4832  Utot 2753363
4833  Utot 2483766
4834  Utot 2747851
4835  Utot 2525806
4836  Utot 2862343
4837  Utot 2718125
4838  Utot 2804525
4839  Utot 2634063
4840  Utot 2678071
4841  Utot 2702394
4842  Utot 2627801
4843  Utot 2685883
4844  Utot 2806429
4845  Utot 2562013
4846  Utot 2736255
4847  Utot 2801963
4848  Utot 2569042
4849  Utot 2784931
4850  Utot 2760237
4851  Utot 2711960
4852  Utot 2710771
4853  Utot 2487914
4854  Utot 2595922
4855  Utot 2657447
4856  Utot 2844217
4857  Utot 2662295
4858  Utot 2742102
4859  Utot 2698022
4860  Utot 2647657
4861  Utot 2603197
4862  Utot 2729298
4863  Utot 2786158
4864  Utot 2521264
4865  Utot 2757347
4866  Utot 2803933
4867  Utot 2818715
4868  Utot 2529910
4869  Utot 2597507
4870  Utot 2835162
4871  Utot 2615522
4872  Utot 2958056
4873  Utot 2712950
4874  Utot 2692817
4875  Utot 2574026
4876  Utot 2730032
4877  Utot 2851192
4878  Utot 2777402
4879  Utot 2771992
4880  Utot 3010593
4881  Utot 2553883
4882  Utot 2566989
4883  Utot 2726533
4884  Utot 2789146
4885  Utot 2712141
4886  Utot 2757720
4887  Utot 2577181
4888  Utot 2654415
4889  Utot 2874467
4890  Utot 2714624
4891  Utot 2815668
4892  Utot 2620090
4893  Utot 2740833
4894  Utot 2844728
4895  Utot 2707627
4896  Utot 2610051
4897  Utot 2576742
4898  Utot 2814453
4899  Utot 2701744
4900  Utot 2779087
4901  Utot 2645625
4902  Utot 2735256
4903  Utot 2571343
4904  Utot 2694545
4905  Utot 2823338
4906  Utot 2844054
4907  Utot 2766214
4908  Utot 2644012
4909  Utot 2730432
4910  Utot 2641064
4911  Utot 2655498
4912  Utot 2506041
4913  Utot 2640930
4914  Utot 2762220
4915  Utot 2745045
4916  Utot 2719463
4917  Utot 2768777
4918  Utot 2745585
4919  Utot 2629881
4920  Utot 2609006
4921  Utot 2884434
4922  Utot 2601542
4923  Utot 2738192
4924  Utot 2851774
4925  Utot 2638948
4926  Utot 2695542
4927  Utot 2766504
4928  Utot 2830551
4929  Utot 2710429
4930  Utot 2669698
4931  Utot 2485347
4932  Utot 2534115
4933  Utot 2822011
4934  Utot 2650181
4935  Utot 2731297
4936  Utot 2746993
4937  Utot 2582250
4938  Utot 2682362
4939  Utot 2829850
4940  Utot 2646878
4941  Utot 2939391
4942  Utot 2592352
4943  Utot 2680720
4944  Utot 2649840
4945  Utot 2586564
4946  Utot 2533446
4947  Utot 2635722
4948  Utot 2860395
4949  Utot 2512661
4950  Utot 2537942
4951  Utot 2781288
4952  Utot 2708071
4953  Utot 2848864
4954  Utot 2657927
4955  Utot 2773064
4956  Utot 2704012
4957  Utot 2853170
4958  Utot 2781848
4959  Utot 2827072
4960  Utot 2478845
4961  Utot 2667515
4962  Utot 2532874
4963  Utot 2805418
4964  Utot 2911299
4965  Utot 2810289
4966  Utot 2869224
4967  Utot 2692572
4968  Utot 2866276
4969  Utot 2864244
4970  Utot 2693465
4971  Utot 2742364
4972  Utot 2709928
4973  Utot 2720425
4974  Utot 2632520
4975  Utot 2715547
4976  Utot 2490158
4977  Utot 2830387
4978  Utot 2755355
4979  Utot 2789121
4980  Utot 2662520
4981  Utot 2827446
4982  Utot 2906417
4983  Utot 2966054
4984  Utot 2888622
4985  Utot 2692740
4986  Utot 2957101
4987  Utot 2692365
4988  Utot 2747467
4989  Utot 2645584
4990  Utot 2579150
4991  Utot 2806410
4992  Utot 2628630
4993  Utot 2797670
4994  Utot 2735089
4995  Utot 2708953
4996  Utot 2851886
4997  Utot 2843873
4998  Utot 2653178
4999  Utot 2757367
5000  Utot 2680628
5001  Utot 2675396
5002  Utot 2733058
5003  Utot 2546841
5004  Utot 2696595
5005  Utot 2596859
5006  Utot 2652086
5007  Utot 2613458
5008  Utot 2524761
5009  Utot 2792088
5010  Utot 2771211
5011  Utot 2662289
5012  Utot 2797419
5013  Utot 2684854
5014  Utot 2786343
5015  Utot 2706930
5016  Utot 2727701
5017  Utot 2659745
5018  Utot 2595218
5019  Utot 2759772
5020  Utot 2737783
5021  Utot 2810459
5022  Utot 2748767
5023  Utot 2824374
5024  Utot 2800073
5025  Utot 2844548
5026  Utot 2708864
5027  Utot 2690936
5028  Utot 2722761
5029  Utot 2809753
5030  Utot 2639760
5031  Utot 2768359
5032  Utot 2664241
5033  Utot 2741359
5034  Utot 2719462
5035  Utot 2615217
5036  Utot 2939431
5037  Utot 2653563
5038  Utot 2615387
5039  Utot 2718932
5040  Utot 2758983
5041  Utot 2732561
5042  Utot 2577816
5043  Utot 2657598
5044  Utot 2633703
5045  Utot 2803923
5046  Utot 2648155
5047  Utot 2683985
5048  Utot 2566219
5049  Utot 2874889
5050  Utot 2746344
5051  Utot 2639330
5052  Utot 2746283
5053  Utot 2545677
5054  Utot 2720948
5055  Utot 2641088
5056  Utot 2734707
5057  Utot 2704532
5058  Utot 2798101
5059  Utot 2802007
5060  Utot 2714615
5061  Utot 2594246
5062  Utot 2711368
5063  Utot 2650525
5064  Utot 2790925
5065  Utot 2602019
5066  Utot 2783712
5067  Utot 2660521
5068  Utot 2723648
5069  Utot 2727118
5070  Utot 2545774
5071  Utot 2709330
5072  Utot 2585800
5073  Utot 2622411
5074  Utot 2668514
5075  Utot 2668981
5076  Utot 2448507
5077  Utot 2633936
5078  Utot 2642986
5079  Utot 2497404
5080  Utot 2566086
5081  Utot 2586739
5082  Utot 2806347
5083  Utot 2787356
5084  Utot 2750563
5085  Utot 2587243
5086  Utot 2623447
5087  Utot 2786154
5088  Utot 2932724
5089  Utot 2728862
5090  Utot 2662318
5091  Utot 2799694
5092  Utot 2814097
5093  Utot 2639105
5094  Utot 2633445
5095  Utot 2732104
5096  Utot 2711124
5097  Utot 2836168
5098  Utot 2759579
5099  Utot 2787042
5100  Utot 2673497
5101  Utot 2625233
5102  Utot 2470443
5103  Utot 2684699
5104  Utot 2738428
5105  Utot 2620887
5106  Utot 2667398
5107  Utot 2643249
5108  Utot 2706418
5109  Utot 2711662
5110  Utot 2685223
5111  Utot 2681429
5112  Utot 2652006
5113  Utot 2727961
5114  Utot 2733489
5115  Utot 2666753
5116  Utot 2766542
5117  Utot 2583957
5118  Utot 2732216
5119  Utot 2755673
5120  Utot 2714765
5121  Utot 2852052
5122  Utot 2744357
5123  Utot 2574724
5124  Utot 2697668
5125  Utot 2733594
5126  Utot 2754275
5127  Utot 2595779
5128  Utot 2796795
5129  Utot 2727617
5130  Utot 2733653
5131  Utot 2516223
5132  Utot 2602482
5133  Utot 2741341
5134  Utot 2533661
5135  Utot 2702713
5136  Utot 2747665
5137  Utot 2655112
5138  Utot 2697003
5139  Utot 2670365
5140  Utot 2765391
5141  Utot 2778727
5142  Utot 2652697
5143  Utot 2712349
5144  Utot 2733199
5145  Utot 2860244
5146  Utot 2787156
5147  Utot 2602563
5148  Utot 2658436
5149  Utot 2711886
5150  Utot 2823882
5151  Utot 2621014
5152  Utot 2775388
5153  Utot 2685084
5154  Utot 2719833
5155  Utot 2584085
5156  Utot 2594714
5157  Utot 2602347
5158  Utot 2429258
5159  Utot 2698051
5160  Utot 2614758
5161  Utot 2710149
5162  Utot 2634461
5163  Utot 2639966
5164  Utot 2973297
5165  Utot 2545605
5166  Utot 2594516
5167  Utot 2705149
5168  Utot 2771718
5169  Utot 2731670
5170  Utot 2693884
5171  Utot 2617383
5172  Utot 2814305
5173  Utot 2691853
5174  Utot 2688569
5175  Utot 2633808
5176  Utot 2967011
5177  Utot 2624203
5178  Utot 2701536
5179  Utot 2671904
5180  Utot 2502174
5181  Utot 2719066
5182  Utot 2874587
5183  Utot 2828385
5184  Utot 2703182
5185  Utot 2866398
5186  Utot 2885631
5187  Utot 2914611
5188  Utot 3036907
5189  Utot 2764454
5190  Utot 2774009
5191  Utot 2744582
5192  Utot 2734794
5193  Utot 2731853
5194  Utot 2763049
5195  Utot 2563146
5196  Utot 2851286
5197  Utot 2616237
5198  Utot 2710496
5199  Utot 2821888
5200  Utot 2537216
5201  Utot 2977014
5202  Utot 2573835
5203  Utot 2672205
5204  Utot 2867131
5205  Utot 2677858
5206  Utot 2757245
5207  Utot 2732467
5208  Utot 2631403
5209  Utot 2753751
5210  Utot 2663802
5211  Utot 2607231
5212  Utot 2752277
5213  Utot 2833752
5214  Utot 2993158
5215  Utot 2973238
5216  Utot 2749957
5217  Utot 2862211
5218  Utot 2663396
5219  Utot 2552596
5220  Utot 2637548
5221  Utot 2507926
5222  Utot 2570363
5223  Utot 2895708
5224  Utot 2834744
5225  Utot 2731904
5226  Utot 2801125
5227  Utot 2985525
5228  Utot 2688029
5229  Utot 2595668
5230  Utot 2757346
5231  Utot 2838080
5232  Utot 2502117
5233  Utot 2644108
5234  Utot 2720455
5235  Utot 2797067
5236  Utot 2611892
5237  Utot 2656580
5238  Utot 2859787
5239  Utot 2524943
5240  Utot 2787830
5241  Utot 2825597
5242  Utot 2751562
5243  Utot 2744388
5244  Utot 2635338
5245  Utot 2792290
5246  Utot 2810036
5247  Utot 2706138
5248  Utot 2662380
5249  Utot 2632476
5250  Utot 2851503
5251  Utot 2752442
5252  Utot 2703995
5253  Utot 3148424
5254  Utot 2522470
5255  Utot 2720759
5256  Utot 2808013
5257  Utot 2694200
5258  Utot 2705318
5259  Utot 2676683
5260  Utot 2775811
5261  Utot 2614708
5262  Utot 2549694
5263  Utot 2860675
5264  Utot 2665607
5265  Utot 2872577
5266  Utot 2523975
5267  Utot 2622216
5268  Utot 2626619
5269  Utot 2606971
5270  Utot 2803584
5271  Utot 2537369
5272  Utot 2676118
5273  Utot 2791317
5274  Utot 2623475
5275  Utot 2645164
5276  Utot 2623354
5277  Utot 2556224
5278  Utot 2656443
5279  Utot 2599749
5280  Utot 2737952
5281  Utot 2731299
5282  Utot 2685767
5283  Utot 2637394
5284  Utot 2887053
5285  Utot 2839037
5286  Utot 2751045
5287  Utot 2959497
5288  Utot 2553743
5289  Utot 2716237
5290  Utot 2530087
5291  Utot 2721567
5292  Utot 2688066
5293  Utot 2600642
5294  Utot 2872496
5295  Utot 2836376
5296  Utot 2638283
5297  Utot 2715386
5298  Utot 2598152
5299  Utot 2779082
5300  Utot 2712209
5301  Utot 2744080
5302  Utot 2543065
5303  Utot 2933083
5304  Utot 2644885
5305  Utot 2763907
5306  Utot 2749115
5307  Utot 2572708
5308  Utot 2759436
5309  Utot 2592992
5310  Utot 2618220
5311  Utot 2804383
5312  Utot 2691332
5313  Utot 2875238
5314  Utot 2466366
5315  Utot 2894485
5316  Utot 2625720
5317  Utot 2746130
5318  Utot 2682079
5319  Utot 2725660
5320  Utot 2729949
5321  Utot 2611315
5322  Utot 2621157
5323  Utot 2714098
5324  Utot 2782738
5325  Utot 2674547
5326  Utot 2663945
5327  Utot 2599251
5328  Utot 2651464
5329  Utot 2669726
5330  Utot 2845658
5331  Utot 2601089
5332  Utot 2680422
5333  Utot 2741693
5334  Utot 2810676
5335  Utot 2621642
5336  Utot 2894869
5337  Utot 2601697
5338  Utot 2659558
5339  Utot 2656279
5340  Utot 2628510
5341  Utot 2756804
5342  Utot 2635634
5343  Utot 2655326
5344  Utot 2683517
5345  Utot 2735196
5346  Utot 2667660
5347  Utot 2746317
5348  Utot 2781700
5349  Utot 2687696
5350  Utot 2621388
5351  Utot 2757228
5352  Utot 2688220
5353  Utot 2731060
5354  Utot 2623969
5355  Utot 2726379
5356  Utot 2728422
5357  Utot 2772373
5358  Utot 2680478
5359  Utot 2726932
5360  Utot 2703974
5361  Utot 2801227
5362  Utot 2648633
5363  Utot 2517237
5364  Utot 2878323
5365  Utot 2445408
5366  Utot 2456419
5367  Utot 2589496
5368  Utot 2813770
5369  Utot 2763459
5370  Utot 2684519
5371  Utot 2593472
5372  Utot 2678316
5373  Utot 2684542
5374  Utot 2837004
5375  Utot 2655061
5376  Utot 2587417
5377  Utot 2636316
5378  Utot 2768472
5379  Utot 2835496
5380  Utot 2580163
5381  Utot 2607215
5382  Utot 2649420
5383  Utot 2793548
5384  Utot 2519503
5385  Utot 2710883
5386  Utot 2880496
5387  Utot 2742498
5388  Utot 2719838
5389  Utot 2751294
5390  Utot 2700886
5391  Utot 2509874
5392  Utot 2891623
5393  Utot 3053199
5394  Utot 2778717
5395  Utot 2786939
5396  Utot 2674049
5397  Utot 2686490
5398  Utot 2658177
5399  Utot 2718833
5400  Utot 2729907
5401  Utot 2896883
5402  Utot 2804973
5403  Utot 2840984
5404  Utot 2723915
5405  Utot 2634027
5406  Utot 2660567
5407  Utot 2827080
5408  Utot 2621533
5409  Utot 2827377
5410  Utot 2616659
5411  Utot 2757834
5412  Utot 2562661
5413  Utot 2647577
5414  Utot 2571766
5415  Utot 2797331
5416  Utot 2554645
5417  Utot 2601833
5418  Utot 2769871
5419  Utot 2586010
5420  Utot 2720176
5421  Utot 2694741
5422  Utot 2733119
5423  Utot 2755331
5424  Utot 2880553
5425  Utot 2775937
5426  Utot 2863366
5427  Utot 2684923
5428  Utot 2729890
5429  Utot 2753682
5430  Utot 2691616
5431  Utot 2812475
5432  Utot 2719349
5433  Utot 2595511
5434  Utot 2646926
5435  Utot 2675917
5436  Utot 2727656
5437  Utot 2516890
5438  Utot 2708073
5439  Utot 2776910
5440  Utot 2568983
5441  Utot 2697465
5442  Utot 2753242
5443  Utot 2722149
5444  Utot 2854797
5445  Utot 2681885
5446  Utot 2662286
5447  Utot 2767751
5448  Utot 2783414
5449  Utot 2765571
5450  Utot 2891564
5451  Utot 2804073
5452  Utot 2689292
5453  Utot 2767975
5454  Utot 2607332
5455  Utot 2578136
5456  Utot 2808089
5457  Utot 2762620
5458  Utot 2832921
5459  Utot 2792300
5460  Utot 2644809
5461  Utot 2693580
5462  Utot 2757604
5463  Utot 2837748
5464  Utot 2787851
5465  Utot 2664497
5466  Utot 2787269
5467  Utot 2861676
5468  Utot 2796026
5469  Utot 2631447
5470  Utot 2743715
5471  Utot 2623561
5472  Utot 2661561
5473  Utot 2690852
5474  Utot 2897202
5475  Utot 2731667
5476  Utot 2753156
5477  Utot 3022642
5478  Utot 2745441
5479  Utot 2471280
5480  Utot 2584787
5481  Utot 2717234
5482  Utot 2775598
5483  Utot 2682496
5484  Utot 2726797
5485  Utot 2808859
5486  Utot 2803585
5487  Utot 2550085
5488  Utot 2605771
5489  Utot 2689918
5490  Utot 2895083
5491  Utot 2641762
5492  Utot 2981759
5493  Utot 2764317
5494  Utot 2746093
5495  Utot 2658885
5496  Utot 2888708
5497  Utot 2703819
5498  Utot 2897447
5499  Utot 2806850
5500  Utot 2557060
5501  Utot 2623681
5502  Utot 2696465
5503  Utot 2570542
5504  Utot 2848113
5505  Utot 2874262
5506  Utot 2763957
5507  Utot 2686165
5508  Utot 2756099
5509  Utot 2682513
5510  Utot 2713746
5511  Utot 2559199
5512  Utot 2751877
5513  Utot 2714410
5514  Utot 2845261
5515  Utot 2753272
5516  Utot 2850011
5517  Utot 2733694
5518  Utot 2797211
5519  Utot 2465778
5520  Utot 2763939
5521  Utot 2664376
5522  Utot 2700636
5523  Utot 2691882
5524  Utot 2674608
5525  Utot 2483737
5526  Utot 2868338
5527  Utot 2709237
5528  Utot 2857782
5529  Utot 2615571
5530  Utot 2531509
5531  Utot 2710764
5532  Utot 2782998
5533  Utot 2926150
5534  Utot 2754676
5535  Utot 2808781
5536  Utot 2582277
5537  Utot 2646136
5538  Utot 2809139
5539  Utot 2853216
5540  Utot 2746019
5541  Utot 2704535
5542  Utot 2568224
5543  Utot 2624665
5544  Utot 2731060
5545  Utot 2691674
5546  Utot 2890307
5547  Utot 2695686
5548  Utot 2925967
5549  Utot 2626723
5550  Utot 2628104
5551  Utot 2935142
5552  Utot 2688391
5553  Utot 2612740
5554  Utot 2844730
5555  Utot 2669724
5556  Utot 2522874
5557  Utot 2761667
5558  Utot 2768535
5559  Utot 2795829
5560  Utot 2819595
5561  Utot 2752658
5562  Utot 2780259
5563  Utot 2762771
5564  Utot 2781942
5565  Utot 2607630
5566  Utot 2550429
5567  Utot 2668575
5568  Utot 2607475
5569  Utot 2773187
5570  Utot 2509616
5571  Utot 2832098
5572  Utot 2691244
5573  Utot 2746114
5574  Utot 2908116
5575  Utot 2681385
5576  Utot 2686371
5577  Utot 2664078
5578  Utot 2773177
5579  Utot 2684510
5580  Utot 2608941
5581  Utot 2688938
5582  Utot 2757072
5583  Utot 2550876
5584  Utot 2707931
5585  Utot 2810123
5586  Utot 2837956
5587  Utot 2658525
5588  Utot 2795680
5589  Utot 2715992
5590  Utot 2957495
5591  Utot 2874906
5592  Utot 2823086
5593  Utot 3024529
5594  Utot 2663958
5595  Utot 2871983
5596  Utot 2701664
5597  Utot 2643398
5598  Utot 2618641
5599  Utot 2632706
5600  Utot 2669623
5601  Utot 2649450
5602  Utot 2800361
5603  Utot 2670539
5604  Utot 2766762
5605  Utot 2624933
5606  Utot 2740342
5607  Utot 2736872
5608  Utot 2556811
5609  Utot 2697666
5610  Utot 2672723
5611  Utot 2638321
5612  Utot 2738622
5613  Utot 2789181
5614  Utot 2609340
5615  Utot 2965786
5616  Utot 2834541
5617  Utot 2631711
5618  Utot 2811008
5619  Utot 2755798
5620  Utot 2574217
5621  Utot 2822174
5622  Utot 2696541
5623  Utot 2708335
5624  Utot 2792030
5625  Utot 2643390
5626  Utot 2861599
5627  Utot 2823422
5628  Utot 2607139
5629  Utot 2770890
5630  Utot 2840249
5631  Utot 2600569
5632  Utot 2649715
5633  Utot 2710758
5634  Utot 2553373
5635  Utot 2652050
5636  Utot 2654557
5637  Utot 2761122
5638  Utot 2835910
5639  Utot 2685328
5640  Utot 2770258
5641  Utot 2753997
5642  Utot 2642830
5643  Utot 2511096
5644  Utot 3015007
5645  Utot 2572904
5646  Utot 2758510
5647  Utot 2627333
5648  Utot 2701840
5649  Utot 2634713
5650  Utot 2697737
5651  Utot 2752531
5652  Utot 2692813
5653  Utot 2777650
5654  Utot 2748306
5655  Utot 2881651
5656  Utot 2667416
5657  Utot 2626772
5658  Utot 2597519
5659  Utot 2724538
5660  Utot 2773571
5661  Utot 2581907
5662  Utot 2696127
5663  Utot 2735594
5664  Utot 2656697
5665  Utot 2712478
5666  Utot 2583618
5667  Utot 2801235
5668  Utot 2631162
5669  Utot 2562956
5670  Utot 2844028
5671  Utot 2534492
5672  Utot 2828447
5673  Utot 2667590
5674  Utot 2947796
5675  Utot 2776001
5676  Utot 2809097
5677  Utot 2721037
5678  Utot 2833228
5679  Utot 2794011
5680  Utot 2951795
5681  Utot 2679942
5682  Utot 2687423
5683  Utot 2622227
5684  Utot 2877315
5685  Utot 2564544
5686  Utot 2636773
5687  Utot 2768592
5688  Utot 2754191
5689  Utot 2805423
5690  Utot 2823598
5691  Utot 2601223
5692  Utot 2824559
5693  Utot 2987252
5694  Utot 2745537
5695  Utot 2846564
5696  Utot 2816765
5697  Utot 2582796
5698  Utot 2641269
5699  Utot 2868002
5700  Utot 2804325
5701  Utot 2780396
5702  Utot 2506759
5703  Utot 2594574
5704  Utot 2566407
5705  Utot 2646229
5706  Utot 2715361
5707  Utot 2786398
5708  Utot 2706628
5709  Utot 2557857
5710  Utot 2769701
5711  Utot 2755335
5712  Utot 2692577
5713  Utot 2677897
5714  Utot 2710352
5715  Utot 2774885
5716  Utot 2767258
5717  Utot 2540577
5718  Utot 2549656
5719  Utot 2759254
5720  Utot 2844510
5721  Utot 2631772
5722  Utot 2875885
5723  Utot 2633275
5724  Utot 2807151
5725  Utot 2656057
5726  Utot 2807436
5727  Utot 2600502
5728  Utot 2669585
5729  Utot 2855471
5730  Utot 2561373
5731  Utot 2916857
5732  Utot 2836627
5733  Utot 2936869
5734  Utot 2762643
5735  Utot 2799061
5736  Utot 2650494
5737  Utot 2848987
5738  Utot 2718009
5739  Utot 2771992
5740  Utot 2640341
5741  Utot 2820604
5742  Utot 2800894
5743  Utot 2526562
5744  Utot 2612556
5745  Utot 2843441
5746  Utot 2546143
5747  Utot 2628224
5748  Utot 2741607
5749  Utot 2589333
5750  Utot 2676333
5751  Utot 2767203
5752  Utot 2879230
5753  Utot 2662175
5754  Utot 2674436
5755  Utot 2833844
5756  Utot 2829123
5757  Utot 2796591
5758  Utot 2680328
5759  Utot 2698753
5760  Utot 2914122
5761  Utot 2813104
5762  Utot 2743056
5763  Utot 2667960
5764  Utot 2507145
5765  Utot 2774670
5766  Utot 2628820
5767  Utot 2626328
5768  Utot 2443387
5769  Utot 2723471
5770  Utot 2615568
5771  Utot 2811285
5772  Utot 2739468
5773  Utot 2658978
5774  Utot 2627247
5775  Utot 2708708
5776  Utot 2632574
5777  Utot 2595763
5778  Utot 2711776
5779  Utot 2566884
5780  Utot 2669193
5781  Utot 2547345
5782  Utot 2957250
5783  Utot 2646682
5784  Utot 2709488
5785  Utot 2938189
5786  Utot 2652075
5787  Utot 2766170
5788  Utot 2616373
5789  Utot 2715453
5790  Utot 2608725
5791  Utot 2759738
5792  Utot 2678671
5793  Utot 2481889
5794  Utot 2620692
5795  Utot 2640847
5796  Utot 2630763
5797  Utot 2637929
5798  Utot 2681577
5799  Utot 2688861
5800  Utot 2580852
5801  Utot 2721692
5802  Utot 2594174
5803  Utot 2746326
5804  Utot 2616919
5805  Utot 2855489
5806  Utot 2665886
5807  Utot 2854168
5808  Utot 2832211
5809  Utot 2699672
5810  Utot 2601929
5811  Utot 2692439
5812  Utot 2773628
5813  Utot 2621067
5814  Utot 2800516
5815  Utot 2829183
5816  Utot 2898535
5817  Utot 2761109
5818  Utot 2877920
5819  Utot 2775882
5820  Utot 2651048
5821  Utot 2833348
5822  Utot 2860670
5823  Utot 2730567
5824  Utot 2667838
5825  Utot 2577141
5826  Utot 2613896
5827  Utot 2644557
5828  Utot 2511099
5829  Utot 2750256
5830  Utot 2810868
5831  Utot 2769013
5832  Utot 2540226
5833  Utot 2816826
5834  Utot 2755761
5835  Utot 2719202
5836  Utot 2594279
5837  Utot 2716002
5838  Utot 2860087
5839  Utot 2583519
5840  Utot 2669107
5841  Utot 2834538
5842  Utot 2874206
5843  Utot 2865933
5844  Utot 2823461
5845  Utot 2672591
5846  Utot 2668410
5847  Utot 2569717
5848  Utot 2740100
5849  Utot 2882452
5850  Utot 2791133
5851  Utot 2731880
5852  Utot 2660243
5853  Utot 2640287
5854  Utot 2779703
5855  Utot 2768754
5856  Utot 2758094
5857  Utot 2725061
5858  Utot 2768012
5859  Utot 2646409
5860  Utot 2643537
5861  Utot 2843722
5862  Utot 2529821
5863  Utot 2705273
5864  Utot 2658095
5865  Utot 2968090
5866  Utot 2887427
5867  Utot 2617853
5868  Utot 2876410
5869  Utot 2762279
5870  Utot 2783579
5871  Utot 2793363
5872  Utot 2661563
5873  Utot 2612241
5874  Utot 2818502
5875  Utot 2745441
5876  Utot 2880584
5877  Utot 2729641
5878  Utot 2705398
5879  Utot 2810558
5880  Utot 2684588
5881  Utot 2814721
5882  Utot 2589228
5883  Utot 2671524
5884  Utot 2715017
5885  Utot 2673684
5886  Utot 2721442
5887  Utot 2628001
5888  Utot 2748128
5889  Utot 2787404
5890  Utot 2720509
5891  Utot 2661126
5892  Utot 2783058
5893  Utot 2802864
5894  Utot 2656264
5895  Utot 2700234
5896  Utot 2700710
5897  Utot 2759034
5898  Utot 2735337
5899  Utot 2781204
5900  Utot 2494406
5901  Utot 2623503
5902  Utot 2546938
5903  Utot 2777956
5904  Utot 2718620
5905  Utot 2723082
5906  Utot 2682422
5907  Utot 2751457
5908  Utot 2775872
5909  Utot 2565417
5910  Utot 2732111
5911  Utot 2649817
5912  Utot 2698684
5913  Utot 2709260
5914  Utot 2629994
5915  Utot 2598215
5916  Utot 2643623
5917  Utot 2769529
5918  Utot 2739340
5919  Utot 2691008
5920  Utot 2696231
5921  Utot 2851709
5922  Utot 2805272
5923  Utot 2911538
5924  Utot 2975963
5925  Utot 2573647
5926  Utot 2867993
5927  Utot 2575628
5928  Utot 2808186
5929  Utot 2872322
5930  Utot 2705820
5931  Utot 2621465
5932  Utot 2839687
5933  Utot 2925559
5934  Utot 2586013
5935  Utot 2717838
5936  Utot 2490631
5937  Utot 2706516
5938  Utot 2660981
5939  Utot 2636615
5940  Utot 2588768
5941  Utot 2709934
5942  Utot 2884903
5943  Utot 2927179
5944  Utot 2687338
5945  Utot 2615445
5946  Utot 2849257
5947  Utot 2588475
5948  Utot 2613904
5949  Utot 2904550
5950  Utot 2555919
5951  Utot 2689659
5952  Utot 2708702
5953  Utot 2694583
5954  Utot 2491457
5955  Utot 2768282
5956  Utot 2906500
5957  Utot 2747697
5958  Utot 2768204
5959  Utot 2700234
5960  Utot 2633497
5961  Utot 2770693
5962  Utot 2672833
5963  Utot 2723042
5964  Utot 2924281
5965  Utot 2863001
5966  Utot 2874627
5967  Utot 2670810
5968  Utot 2719862
5969  Utot 2785003
5970  Utot 2540306
5971  Utot 2783426
5972  Utot 2669585
5973  Utot 2568397
5974  Utot 2792389
5975  Utot 2772962
5976  Utot 2699913
5977  Utot 2779627
5978  Utot 2935339
5979  Utot 2621031
5980  Utot 2663432
5981  Utot 2926267
5982  Utot 2793546
5983  Utot 2794422
5984  Utot 2585590
5985  Utot 2627374
5986  Utot 2747314
5987  Utot 2833193
5988  Utot 2765407
5989  Utot 2666963
5990  Utot 2588557
5991  Utot 2505936
5992  Utot 2604114
5993  Utot 2803598
5994  Utot 2836709
5995  Utot 2657784
5996  Utot 2662449
5997  Utot 2651585
5998  Utot 2688878
5999  Utot 2532054
6000  Utot 2913027
6001  Ntot 2818504
6002  Ntot 2803520
6003  Ntot 2734225
6004  Ntot 2585149
6005  Ntot 2708289
6006  Ntot 2871926
6007  Ntot 2816506
6008  Ntot 2762413
6009  Ntot 2514710
6010  Ntot 2528273
6011  Ntot 2796424
6012  Ntot 2835261
6013  Ntot 2623847
6014  Ntot 2942375
6015  Ntot 2895388
6016  Ntot 2843503
6017  Ntot 2762821
6018  Ntot 2869393
6019  Ntot 2926548
6020  Ntot 2824462
6021  Ntot 2796629
6022  Ntot 2770412
6023  Ntot 2734882
6024  Ntot 2785120
6025  Ntot 2802230
6026  Ntot 2687866
6027  Ntot 2657488
6028  Ntot 2848014
6029  Ntot 2713703
6030  Ntot 2734892
6031  Ntot 2780026
6032  Ntot 2770790
6033  Ntot 2854489
6034  Ntot 2705212
6035  Ntot 2625939
6036  Ntot 2809443
6037  Ntot 2859155
6038  Ntot 2774719
6039  Ntot 2949221
6040  Ntot 2669491
6041  Ntot 2959830
6042  Ntot 2865597
6043  Ntot 2817691
6044  Ntot 2631133
6045  Ntot 2773662
6046  Ntot 2961428
6047  Ntot 2543012
6048  Ntot 2689375
6049  Ntot 2700642
6050  Ntot 2653869
6051  Ntot 2755666
6052  Ntot 2508602
6053  Ntot 2767413
6054  Ntot 2844307
6055  Ntot 2853899
6056  Ntot 3015758
6057  Ntot 2873995
6058  Ntot 2735807
6059  Ntot 2831263
6060  Ntot 2910642
6061  Ntot 2705843
6062  Ntot 2520667
6063  Ntot 2829352
6064  Ntot 2648458
6065  Ntot 2807023
6066  Ntot 2787346
6067  Ntot 2727563
6068  Ntot 2841917
6069  Ntot 2719943
6070  Ntot 2699398
6071  Ntot 2790487
6072  Ntot 2881089
6073  Ntot 2849091
6074  Ntot 2901871
6075  Ntot 2853538
6076  Ntot 2601957
6077  Ntot 2643708
6078  Ntot 2808964
6079  Ntot 2814278
6080  Ntot 2904527
6081  Ntot 2757282
6082  Ntot 2843489
6083  Ntot 2680370
6084  Ntot 2781945
6085  Ntot 2861692
6086  Ntot 2624852
6087  Ntot 2820815
6088  Ntot 2828404
6089  Ntot 2650378
6090  Ntot 2740269
6091  Ntot 2745685
6092  Ntot 2875332
6093  Ntot 2670016
6094  Ntot 2876159
6095  Ntot 2864742
6096  Ntot 2724295
6097  Ntot 2876460
6098  Ntot 2598522
6099  Ntot 2737696
6100  Ntot 2779794
6101  Ntot 2787618
6102  Ntot 2936920
6103  Ntot 2655143
6104  Ntot 2778344
6105  Ntot 2685599
6106  Ntot 2750149
6107  Ntot 2906786
6108  Ntot 2680333
6109  Ntot 2871820
6110  Ntot 2725537
6111  Ntot 2809534
6112  Ntot 2835257
6113  Ntot 2635709
6114  Ntot 2720595
6115  Ntot 2795574
6116  Ntot 2680221
6117  Ntot 2504376
6118  Ntot 2771722
6119  Ntot 2666125
6120  Ntot 2854842
6121  Ntot 2738761
6122  Ntot 2649712
6123  Ntot 2759683
6124  Ntot 2865638
6125  Ntot 2670673
6126  Ntot 2864764
6127  Ntot 2845522
6128  Ntot 2770076
6129  Ntot 2689113
6130  Ntot 2625977
6131  Ntot 2713019
6132  Ntot 2598302
6133  Ntot 2944630
6134  Ntot 2809478
6135  Ntot 2922897
6136  Ntot 2690468
6137  Ntot 2795320
6138  Ntot 2727181
6139  Ntot 2783291
6140  Ntot 2687036
6141  Ntot 2775750
6142  Ntot 2724802
6143  Ntot 2667988
6144  Ntot 2784797
6145  Ntot 2807183
6146  Ntot 2709664
6147  Ntot 2672312
6148  Ntot 2684252
6149  Ntot 2882497
6150  Ntot 2730317
6151  Ntot 2823545
6152  Ntot 2603002
6153  Ntot 2873054
6154  Ntot 2668528
6155  Ntot 2755064
6156  Ntot 2853635
6157  Ntot 2592847
6158  Ntot 2694950
6159  Ntot 2748479
6160  Ntot 2674640
6161  Ntot 2668683
6162  Ntot 2660100
6163  Ntot 2683179
6164  Ntot 2826551
6165  Ntot 2513969
6166  Ntot 2849092
6167  Ntot 2638633
6168  Ntot 2771692
6169  Ntot 2700347
6170  Ntot 2686415
6171  Ntot 2837876
6172  Ntot 2784153
6173  Ntot 2760957
6174  Ntot 2935252
6175  Ntot 2614677
6176  Ntot 2857894
6177  Ntot 2664785
6178  Ntot 2765481
6179  Ntot 2625623
6180  Ntot 2649606
6181  Ntot 2620090
6182  Ntot 2743574
6183  Ntot 2908420
6184  Ntot 2809310
6185  Ntot 2581830
6186  Ntot 2919489
6187  Ntot 2730750
6188  Ntot 2683799
6189  Ntot 2890007
6190  Ntot 2726140
6191  Ntot 2715000
6192  Ntot 2626972
6193  Ntot 2641213
6194  Ntot 2879278
6195  Ntot 2744836
6196  Ntot 2871207
6197  Ntot 2626782
6198  Ntot 2708013
6199  Ntot 2863372
6200  Ntot 2781345
6201  Ntot 2789167
6202  Ntot 2647954
6203  Ntot 2674049
6204  Ntot 2763492
6205  Ntot 2842076
6206  Ntot 2750855
6207  Ntot 2754022
6208  Ntot 2579759
6209  Ntot 2661004
6210  Ntot 2475435
6211  Ntot 2640538
6212  Ntot 2713159
6213  Ntot 2816581
6214  Ntot 2773261
6215  Ntot 2565332
6216  Ntot 2625555
6217  Ntot 2702721
6218  Ntot 2691654
6219  Ntot 2708713
6220  Ntot 2762770
6221  Ntot 2828239
6222  Ntot 2662489
6223  Ntot 2610096
6224  Ntot 2781385
6225  Ntot 2880984
6226  Ntot 2670304
6227  Ntot 2717867
6228  Ntot 2803525
6229  Ntot 2763066
6230  Ntot 2897972
6231  Ntot 2867519
6232  Ntot 2823856
6233  Ntot 2607798
6234  Ntot 2668407
6235  Ntot 2892477
6236  Ntot 2832274
6237  Ntot 2854706
6238  Ntot 2759090
6239  Ntot 2658628
6240  Ntot 2775330
6241  Ntot 2749674
6242  Ntot 2910350
6243  Ntot 2754142
6244  Ntot 2794477
6245  Ntot 2870695
6246  Ntot 2873719
6247  Ntot 2705687
6248  Ntot 2809772
6249  Ntot 2836781
6250  Ntot 2538824
6251  Ntot 2747176
6252  Ntot 2748245
6253  Ntot 2750170
6254  Ntot 2683523
6255  Ntot 2932883
6256  Ntot 2706576
6257  Ntot 2754825
6258  Ntot 2716348
6259  Ntot 2747711
6260  Ntot 2599961
6261  Ntot 2812274
6262  Ntot 2675724
6263  Ntot 2840989
6264  Ntot 2727154
6265  Ntot 2801603
6266  Ntot 2727825
6267  Ntot 2760606
6268  Ntot 3088118
6269  Ntot 2709710
6270  Ntot 2791661
6271  Ntot 2603433
6272  Ntot 2657316
6273  Ntot 2718281
6274  Ntot 2678124
6275  Ntot 2752412
6276  Ntot 2731415
6277  Ntot 2763699
6278  Ntot 2694213
6279  Ntot 2894491
6280  Ntot 2606161
6281  Ntot 2933686
6282  Ntot 2665498
6283  Ntot 2729763
6284  Ntot 2869339
6285  Ntot 2888588
6286  Ntot 2613590
6287  Ntot 2768613
6288  Ntot 2788426
6289  Ntot 2816789
6290  Ntot 2816780
6291  Ntot 2864152
6292  Ntot 2977362
6293  Ntot 2855842
6294  Ntot 2575396
6295  Ntot 2779169
6296  Ntot 2762818
6297  Ntot 2642594
6298  Ntot 2742383
6299  Ntot 2832907
6300  Ntot 2810874
6301  Ntot 2556614
6302  Ntot 2851641
6303  Ntot 2840718
6304  Ntot 2813427
6305  Ntot 2840239
6306  Ntot 2915306
6307  Ntot 2807329
6308  Ntot 2833048
6309  Ntot 2541531
6310  Ntot 2978410
6311  Ntot 2844330
6312  Ntot 2577429
6313  Ntot 2900548
6314  Ntot 2771900
6315  Ntot 2710718
6316  Ntot 2740649
6317  Ntot 2794501
6318  Ntot 3093187
6319  Ntot 2776216
6320  Ntot 2677309
6321  Ntot 2946432
6322  Ntot 2770078
6323  Ntot 2781197
6324  Ntot 3034501
6325  Ntot 2741341
6326  Ntot 2879597
6327  Ntot 2610508
6328  Ntot 2657502
6329  Ntot 2694594
6330  Ntot 2955392
6331  Ntot 2943540
6332  Ntot 2826395
6333  Ntot 2876526
6334  Ntot 2738679
6335  Ntot 2692948
6336  Ntot 2779071
6337  Ntot 2712069
6338  Ntot 2830176
6339  Ntot 2736903
6340  Ntot 2986270
6341  Ntot 2544552
6342  Ntot 2637869
6343  Ntot 2757521
6344  Ntot 2576605
6345  Ntot 2827835
6346  Ntot 2716441
6347  Ntot 2924567
6348  Ntot 2991387
6349  Ntot 2794880
6350  Ntot 3026810
6351  Ntot 2767756
6352  Ntot 2896227
6353  Ntot 2793038
6354  Ntot 2994545
6355  Ntot 2769896
6356  Ntot 2760865
6357  Ntot 2775000
6358  Ntot 2754383
6359  Ntot 2990402
6360  Ntot 2772673
6361  Ntot 2709070
6362  Ntot 2774599
6363  Ntot 2743888
6364  Ntot 2746802
6365  Ntot 2800083
6366  Ntot 2747956
6367  Ntot 2752925
6368  Ntot 2796565
6369  Ntot 2586060
6370  Ntot 2930925
6371  Ntot 3013306
6372  Ntot 2873803
6373  Ntot 2779298
6374  Ntot 2647766
6375  Ntot 2872833
6376  Ntot 2597525
6377  Ntot 2569074
6378  Ntot 2783752
6379  Ntot 2829736
6380  Ntot 2686669
6381  Ntot 2846576
6382  Ntot 2690574
6383  Ntot 2923858
6384  Ntot 2769567
6385  Ntot 2879578
6386  Ntot 2798098
6387  Ntot 2761407
6388  Ntot 2787233
6389  Ntot 2959268
6390  Ntot 2742579
6391  Ntot 2699673
6392  Ntot 2677581
6393  Ntot 2866555
6394  Ntot 2823468
6395  Ntot 2645461
6396  Ntot 2581263
6397  Ntot 2773654
6398  Ntot 2896920
6399  Ntot 2747275
6400  Ntot 2672025
6401  Ntot 2756951
6402  Ntot 2644427
6403  Ntot 2757859
6404  Ntot 2853448
6405  Ntot 2781450
6406  Ntot 2775031
6407  Ntot 2581669
6408  Ntot 2662551
6409  Ntot 2791101
6410  Ntot 2603634
6411  Ntot 2805355
6412  Ntot 2642094
6413  Ntot 2694856
6414  Ntot 2831684
6415  Ntot 2674803
6416  Ntot 2962421
6417  Ntot 2638812
6418  Ntot 2804359
6419  Ntot 2837154
6420  Ntot 2751971
6421  Ntot 2734357
6422  Ntot 2778230
6423  Ntot 2597422
6424  Ntot 2687528
6425  Ntot 2773364
6426  Ntot 2693157
6427  Ntot 2747038
6428  Ntot 2653218
6429  Ntot 2665293
6430  Ntot 2737304
6431  Ntot 2760736
6432  Ntot 2804660
6433  Ntot 2724152
6434  Ntot 2754617
6435  Ntot 2778915
6436  Ntot 2775921
6437  Ntot 2648192
6438  Ntot 2747278
6439  Ntot 2598651
6440  Ntot 2808963
6441  Ntot 2848147
6442  Ntot 2772652
6443  Ntot 2781975
6444  Ntot 2540505
6445  Ntot 2797754
6446  Ntot 2775407
6447  Ntot 2789983
6448  Ntot 2722463
6449  Ntot 2933286
6450  Ntot 2652222
6451  Ntot 2690166
6452  Ntot 2791656
6453  Ntot 2880067
6454  Ntot 2673779
6455  Ntot 2904499
6456  Ntot 2709655
6457  Ntot 2738891
6458  Ntot 2727857
6459  Ntot 2693053
6460  Ntot 3066683
6461  Ntot 2827974
6462  Ntot 2622678
6463  Ntot 2870783
6464  Ntot 2699440
6465  Ntot 2742024
6466  Ntot 2781336
6467  Ntot 2781415
6468  Ntot 2809973
6469  Ntot 2508254
6470  Ntot 2973583
6471  Ntot 2622832
6472  Ntot 2828322
6473  Ntot 3036417
6474  Ntot 2612052
6475  Ntot 2842712
6476  Ntot 2723267
6477  Ntot 2650888
6478  Ntot 2893756
6479  Ntot 2623125
6480  Ntot 2674972
6481  Ntot 2766048
6482  Ntot 2646323
6483  Ntot 2676436
6484  Ntot 2698665
6485  Ntot 2932562
6486  Ntot 2844258
6487  Ntot 2640573
6488  Ntot 2654434
6489  Ntot 2698554
6490  Ntot 2686706
6491  Ntot 2883820
6492  Ntot 2745980
6493  Ntot 2816821
6494  Ntot 2719470
6495  Ntot 2751767
6496  Ntot 2959126
6497  Ntot 2810448
6498  Ntot 2878609
6499  Ntot 2819350
6500  Ntot 2720154
6501  Ntot 2796809
6502  Ntot 2573008
6503  Ntot 2760718
6504  Ntot 2742230
6505  Ntot 2774536
6506  Ntot 3023684
6507  Ntot 2869381
6508  Ntot 2751500
6509  Ntot 2839967
6510  Ntot 2716849
6511  Ntot 2758806
6512  Ntot 2760349
6513  Ntot 2782893
6514  Ntot 2724969
6515  Ntot 2917836
6516  Ntot 2590334
6517  Ntot 2543091
6518  Ntot 2841422
6519  Ntot 2760415
6520  Ntot 2750730
6521  Ntot 2795557
6522  Ntot 2627168
6523  Ntot 2824297
6524  Ntot 2689359
6525  Ntot 2779647
6526  Ntot 2723545
6527  Ntot 2761664
6528  Ntot 2703407
6529  Ntot 3149259
6530  Ntot 2856180
6531  Ntot 2914652
6532  Ntot 2796445
6533  Ntot 2791243
6534  Ntot 2729253
6535  Ntot 2657674
6536  Ntot 2698871
6537  Ntot 2797823
6538  Ntot 2729172
6539  Ntot 2806894
6540  Ntot 2829694
6541  Ntot 2825446
6542  Ntot 2722799
6543  Ntot 2825625
6544  Ntot 2623874
6545  Ntot 2945523
6546  Ntot 2643424
6547  Ntot 2846837
6548  Ntot 2766410
6549  Ntot 2704978
6550  Ntot 2615914
6551  Ntot 2756712
6552  Ntot 2754305
6553  Ntot 2721385
6554  Ntot 2840181
6555  Ntot 2654000
6556  Ntot 2701636
6557  Ntot 2684650
6558  Ntot 2660772
6559  Ntot 2731017
6560  Ntot 2683721
6561  Ntot 2694330
6562  Ntot 2702394
6563  Ntot 2946522
6564  Ntot 2831316
6565  Ntot 2612874
6566  Ntot 2777996
6567  Ntot 2699178
6568  Ntot 2741066
6569  Ntot 2746698
6570  Ntot 2703113
6571  Ntot 2684063
6572  Ntot 2803708
6573  Ntot 2824416
6574  Ntot 2787388
6575  Ntot 2753047
6576  Ntot 2774335
6577  Ntot 2728515
6578  Ntot 2763185
6579  Ntot 2688214
6580  Ntot 2728068
6581  Ntot 2810885
6582  Ntot 2720370
6583  Ntot 2792330
6584  Ntot 2561722
6585  Ntot 2907580
6586  Ntot 2686273
6587  Ntot 2782132
6588  Ntot 2630225
6589  Ntot 2716640
6590  Ntot 2826815
6591  Ntot 2690216
6592  Ntot 2774912
6593  Ntot 2674467
6594  Ntot 2592568
6595  Ntot 2854653
6596  Ntot 2771179
6597  Ntot 2777574
6598  Ntot 2802956
6599  Ntot 2769580
6600  Ntot 2710185
6601  Ntot 2761549
6602  Ntot 3029625
6603  Ntot 2803666
6604  Ntot 2710911
6605  Ntot 2744394
6606  Ntot 2751047
6607  Ntot 2783236
6608  Ntot 2925873
6609  Ntot 2730002
6610  Ntot 2801224
6611  Ntot 2721768
6612  Ntot 2677898
6613  Ntot 2735052
6614  Ntot 2628905
6615  Ntot 2731994
6616  Ntot 2787319
6617  Ntot 2890531
6618  Ntot 2901350
6619  Ntot 2742312
6620  Ntot 2727162
6621  Ntot 2641389
6622  Ntot 2875120
6623  Ntot 2873257
6624  Ntot 2566276
6625  Ntot 2607487
6626  Ntot 2734676
6627  Ntot 2698015
6628  Ntot 2737050
6629  Ntot 2808899
6630  Ntot 2525411
6631  Ntot 2760276
6632  Ntot 2840553
6633  Ntot 2778839
6634  Ntot 2724696
6635  Ntot 2690451
6636  Ntot 2832220
6637  Ntot 2760721
6638  Ntot 2726035
6639  Ntot 2675605
6640  Ntot 2892386
6641  Ntot 2797896
6642  Ntot 2808611
6643  Ntot 2998598
6644  Ntot 2719384
6645  Ntot 2682467
6646  Ntot 2636598
6647  Ntot 2766403
6648  Ntot 2729953
6649  Ntot 2935124
6650  Ntot 2651369
6651  Ntot 2610497
6652  Ntot 2695656
6653  Ntot 2780158
6654  Ntot 2847685
6655  Ntot 2770084
6656  Ntot 2475521
6657  Ntot 2870356
6658  Ntot 2658975
6659  Ntot 2764528
6660  Ntot 2853611
6661  Ntot 2789271
6662  Ntot 2737952
6663  Ntot 2755838
6664  Ntot 2827913
6665  Ntot 2792543
6666  Ntot 2791382
6667  Ntot 2710405
6668  Ntot 2614031
6669  Ntot 2767474
6670  Ntot 2738898
6671  Ntot 2768107
6672  Ntot 2796733
6673  Ntot 2873769
6674  Ntot 2751452
6675  Ntot 2862917
6676  Ntot 2615337
6677  Ntot 2683194
6678  Ntot 2755121
6679  Ntot 2832412
6680  Ntot 2627534
6681  Ntot 2767615
6682  Ntot 2704589
6683  Ntot 2949963
6684  Ntot 2783944
6685  Ntot 2611703
6686  Ntot 2956833
6687  Ntot 2909785
6688  Ntot 2761825
6689  Ntot 2785598
6690  Ntot 2871795
6691  Ntot 2609848
6692  Ntot 2786068
6693  Ntot 2857405
6694  Ntot 2764031
6695  Ntot 2781875
6696  Ntot 2662032
6697  Ntot 2761724
6698  Ntot 2772328
6699  Ntot 2685652
6700  Ntot 2663955
6701  Ntot 2694912
6702  Ntot 2671118
6703  Ntot 2638870
6704  Ntot 2769555
6705  Ntot 2834262
6706  Ntot 2695541
6707  Ntot 2722575
6708  Ntot 2870820
6709  Ntot 2777788
6710  Ntot 2834033
6711  Ntot 2774547
6712  Ntot 2656333
6713  Ntot 2665023
6714  Ntot 2805607
6715  Ntot 2816581
6716  Ntot 2639770
6717  Ntot 2853979
6718  Ntot 2720060
6719  Ntot 2810319
6720  Ntot 2648440
6721  Ntot 2770957
6722  Ntot 2482607
6723  Ntot 2692308
6724  Ntot 2800875
6725  Ntot 2682645
6726  Ntot 2706868
6727  Ntot 2717155
6728  Ntot 2638783
6729  Ntot 2629201
6730  Ntot 2757544
6731  Ntot 2848913
6732  Ntot 2799373
6733  Ntot 2565796
6734  Ntot 2607108
6735  Ntot 2643876
6736  Ntot 2574936
6737  Ntot 2772009
6738  Ntot 2821803
6739  Ntot 2691875
6740  Ntot 2640152
6741  Ntot 2667906
6742  Ntot 2796305
6743  Ntot 2561398
6744  Ntot 2568282
6745  Ntot 2662898
6746  Ntot 2636540
6747  Ntot 2588671
6748  Ntot 2849368
6749  Ntot 2851353
6750  Ntot 2698863
6751  Ntot 2734781
6752  Ntot 2709044
6753  Ntot 2733463
6754  Ntot 2832701
6755  Ntot 2757010
6756  Ntot 2723700
6757  Ntot 2818915
6758  Ntot 2670542
6759  Ntot 2841069
6760  Ntot 2752241
6761  Ntot 2726290
6762  Ntot 2675170
6763  Ntot 2908853
6764  Ntot 2654636
6765  Ntot 2677958
6766  Ntot 2639745
6767  Ntot 2839110
6768  Ntot 2767434
6769  Ntot 2826925
6770  Ntot 2896550
6771  Ntot 2655651
6772  Ntot 2715801
6773  Ntot 2838223
6774  Ntot 3015755
6775  Ntot 2680737
6776  Ntot 2596297
6777  Ntot 2698638
6778  Ntot 2594352
6779  Ntot 2640429
6780  Ntot 2594013
6781  Ntot 2657721
6782  Ntot 2738066
6783  Ntot 2707361
6784  Ntot 2756757
6785  Ntot 2739379
6786  Ntot 2649971
6787  Ntot 2858224
6788  Ntot 2688136
6789  Ntot 2722280
6790  Ntot 2827723
6791  Ntot 2693536
6792  Ntot 2799472
6793  Ntot 2788459
6794  Ntot 2751058
6795  Ntot 2711255
6796  Ntot 2641443
6797  Ntot 2720326
6798  Ntot 2796139
6799  Ntot 2788224
6800  Ntot 2738790
6801  Ntot 2812304
6802  Ntot 2785261
6803  Ntot 2708881
6804  Ntot 2843427
6805  Ntot 2814228
6806  Ntot 2672738
6807  Ntot 2881914
6808  Ntot 2754864
6809  Ntot 2738798
6810  Ntot 2839736
6811  Ntot 2650766
6812  Ntot 2592898
6813  Ntot 2852103
6814  Ntot 2660218
6815  Ntot 2671689
6816  Ntot 2661747
6817  Ntot 2762876
6818  Ntot 2736874
6819  Ntot 2657431
6820  Ntot 2722359
6821  Ntot 2906015
6822  Ntot 2639618
6823  Ntot 2570119
6824  Ntot 2840881
6825  Ntot 2838855
6826  Ntot 2707207
6827  Ntot 2911812
6828  Ntot 2774127
6829  Ntot 2708037
6830  Ntot 3064559
6831  Ntot 2873464
6832  Ntot 2737133
6833  Ntot 2675199
6834  Ntot 2849964
6835  Ntot 2744530
6836  Ntot 2798009
6837  Ntot 2591110
6838  Ntot 2727557
6839  Ntot 2627907
6840  Ntot 2554206
6841  Ntot 2817041
6842  Ntot 2649734
6843  Ntot 2719787
6844  Ntot 2696425
6845  Ntot 2724550
6846  Ntot 2806347
6847  Ntot 2760168
6848  Ntot 2707275
6849  Ntot 2716397
6850  Ntot 2731319
6851  Ntot 2723219
6852  Ntot 3070433
6853  Ntot 2640169
6854  Ntot 2645162
6855  Ntot 2728734
6856  Ntot 2667179
6857  Ntot 2776373
6858  Ntot 2705395
6859  Ntot 2943000
6860  Ntot 2723266
6861  Ntot 2797191
6862  Ntot 2768829
6863  Ntot 2632278
6864  Ntot 2745607
6865  Ntot 2804277
6866  Ntot 2760920
6867  Ntot 2704017
6868  Ntot 2745164
6869  Ntot 2807476
6870  Ntot 2868362
6871  Ntot 2695929
6872  Ntot 2793294
6873  Ntot 2754188
6874  Ntot 2727241
6875  Ntot 2662223
6876  Ntot 2724458
6877  Ntot 2627518
6878  Ntot 2866641
6879  Ntot 2792379
6880  Ntot 2693431
6881  Ntot 2700631
6882  Ntot 2730659
6883  Ntot 2658599
6884  Ntot 2928186
6885  Ntot 2642918
6886  Ntot 2756612
6887  Ntot 2748835
6888  Ntot 2666933
6889  Ntot 2871759
6890  Ntot 2806782
6891  Ntot 2568054
6892  Ntot 2645428
6893  Ntot 2669253
6894  Ntot 2698269
6895  Ntot 2768401
6896  Ntot 2826378
6897  Ntot 2634558
6898  Ntot 2607469
6899  Ntot 2794774
6900  Ntot 2858966
6901  Ntot 2785421
6902  Ntot 2727909
6903  Ntot 2732302
6904  Ntot 2809941
6905  Ntot 2593316
6906  Ntot 2706875
6907  Ntot 2829268
6908  Ntot 2646565
6909  Ntot 2758141
6910  Ntot 2831341
6911  Ntot 2677843
6912  Ntot 2714661
6913  Ntot 2858516
6914  Ntot 2950484
6915  Ntot 2621125
6916  Ntot 2747678
6917  Ntot 2738554
6918  Ntot 2848108
6919  Ntot 2641780
6920  Ntot 2673674
6921  Ntot 2882346
6922  Ntot 2725751
6923  Ntot 2687605
6924  Ntot 2778528
6925  Ntot 2670298
6926  Ntot 2690529
6927  Ntot 2839146
6928  Ntot 2750995
6929  Ntot 2876006
6930  Ntot 2819269
6931  Ntot 2758923
6932  Ntot 2666837
6933  Ntot 2794156
6934  Ntot 2660443
6935  Ntot 2646765
6936  Ntot 2869546
6937  Ntot 2675411
6938  Ntot 2696565
6939  Ntot 2742468
6940  Ntot 2702445
6941  Ntot 2624540
6942  Ntot 2824989
6943  Ntot 2710450
6944  Ntot 2919628
6945  Ntot 2806072
6946  Ntot 2598058
6947  Ntot 2640661
6948  Ntot 2654008
6949  Ntot 2700921
6950  Ntot 2618857
6951  Ntot 2680251
6952  Ntot 2628152
6953  Ntot 2695086
6954  Ntot 2661283
6955  Ntot 2667360
6956  Ntot 2669618
6957  Ntot 2825135
6958  Ntot 2748613
6959  Ntot 2714404
6960  Ntot 2879388
6961  Ntot 2808499
6962  Ntot 2720289
6963  Ntot 2761492
6964  Ntot 2726467
6965  Ntot 2915672
6966  Ntot 2732472
6967  Ntot 2767807
6968  Ntot 2812611
6969  Ntot 2876368
6970  Ntot 2750246
6971  Ntot 2846891
6972  Ntot 2729917
6973  Ntot 2762043
6974  Ntot 2779980
6975  Ntot 2617874
6976  Ntot 2748624
6977  Ntot 2706840
6978  Ntot 2707360
6979  Ntot 2635290
6980  Ntot 2754388
6981  Ntot 2769269
6982  Ntot 2665309
6983  Ntot 2869980
6984  Ntot 2770399
6985  Ntot 2790971
6986  Ntot 2657750
6987  Ntot 2872266
6988  Ntot 2608343
6989  Ntot 2719652
6990  Ntot 2648533
6991  Ntot 2955742
6992  Ntot 2916706
6993  Ntot 2725850
6994  Ntot 2645184
6995  Ntot 2891799
6996  Ntot 2754350
6997  Ntot 2632446
6998  Ntot 2588711
6999  Ntot 2818104
7000  Ntot 2656171
7001  Ntot 2817692
7002  Ntot 2941052
7003  Ntot 2799997
7004  Ntot 2752003
7005  Ntot 2725105
7006  Ntot 2809622
7007  Ntot 2748485
7008  Ntot 2740897
7009  Ntot 2601152
7010  Ntot 2902440
7011  Ntot 2649918
7012  Ntot 2797363
7013  Ntot 2752404
7014  Ntot 2822778
7015  Ntot 2650622
7016  Ntot 2710696
7017  Ntot 2783611
7018  Ntot 2772237
7019  Ntot 2716729
7020  Ntot 2729207
7021  Ntot 2838831
7022  Ntot 2629813
7023  Ntot 2663352
7024  Ntot 2745923
7025  Ntot 2797802
7026  Ntot 2816063
7027  Ntot 2914242
7028  Ntot 2865929
7029  Ntot 2771085
7030  Ntot 2734309
7031  Ntot 2674999
7032  Ntot 2747374
7033  Ntot 2713572
7034  Ntot 2675666
7035  Ntot 3034905
7036  Ntot 2677487
7037  Ntot 2624995
7038  Ntot 2548140
7039  Ntot 2760698
7040  Ntot 2939380
7041  Ntot 2803760
7042  Ntot 2924803
7043  Ntot 2634406
7044  Ntot 2813324
7045  Ntot 2792304
7046  Ntot 2894221
7047  Ntot 2665410
7048  Ntot 2706399
7049  Ntot 2620696
7050  Ntot 2809719
7051  Ntot 2787712
7052  Ntot 2702388
7053  Ntot 2804925
7054  Ntot 2680989
7055  Ntot 2876178
7056  Ntot 2748467
7057  Ntot 2887145
7058  Ntot 2952411
7059  Ntot 2677601
7060  Ntot 2825653
7061  Ntot 2819600
7062  Ntot 2816051
7063  Ntot 2742877
7064  Ntot 2571361
7065  Ntot 2774047
7066  Ntot 2711542
7067  Ntot 2805930
7068  Ntot 2798861
7069  Ntot 2672515
7070  Ntot 2675244
7071  Ntot 2807547
7072  Ntot 2616715
7073  Ntot 2578615
7074  Ntot 2613865
7075  Ntot 2706244
7076  Ntot 2830307
7077  Ntot 2747873
7078  Ntot 2872148
7079  Ntot 2739451
7080  Ntot 2621075
7081  Ntot 2754729
7082  Ntot 2723910
7083  Ntot 2675900
7084  Ntot 2824914
7085  Ntot 2944694
7086  Ntot 2809527
7087  Ntot 2756011
7088  Ntot 2631799
7089  Ntot 2719175
7090  Ntot 2889733
7091  Ntot 2577758
7092  Ntot 2747207
7093  Ntot 2643968
7094  Ntot 2714943
7095  Ntot 2615111
7096  Ntot 2606898
7097  Ntot 2770326
7098  Ntot 2596584
7099  Ntot 2903785
7100  Ntot 2684364
7101  Ntot 2905708
7102  Ntot 2647020
7103  Ntot 2689004
7104  Ntot 2914018
7105  Ntot 2715363
7106  Ntot 2651868
7107  Ntot 2737334
7108  Ntot 2716723
7109  Ntot 2743301
7110  Ntot 2686754
7111  Ntot 2732343
7112  Ntot 2835318
7113  Ntot 2624409
7114  Ntot 2696446
7115  Ntot 2646348
7116  Ntot 2741109
7117  Ntot 2849084
7118  Ntot 2704142
7119  Ntot 2795785
7120  Ntot 2837699
7121  Ntot 2732129
7122  Ntot 2732901
7123  Ntot 2819432
7124  Ntot 2826803
7125  Ntot 2602906
7126  Ntot 2588915
7127  Ntot 2771170
7128  Ntot 2888643
7129  Ntot 2860319
7130  Ntot 2859584
7131  Ntot 2857534
7132  Ntot 2788180
7133  Ntot 2654128
7134  Ntot 2896397
7135  Ntot 2656617
7136  Ntot 2683331
7137  Ntot 2838459
7138  Ntot 2843784
7139  Ntot 2766658
7140  Ntot 2831535
7141  Ntot 2889274
7142  Ntot 2814735
7143  Ntot 2820229
7144  Ntot 2706373
7145  Ntot 2714778
7146  Ntot 2728571
7147  Ntot 2707561
7148  Ntot 2607934
7149  Ntot 2601284
7150  Ntot 2601693
7151  Ntot 2796294
7152  Ntot 2895951
7153  Ntot 2800508
7154  Ntot 2690426
7155  Ntot 2891761
7156  Ntot 2677395
7157  Ntot 2519033
7158  Ntot 3017905
7159  Ntot 2533263
7160  Ntot 2701891
7161  Ntot 2614773
7162  Ntot 2769192
7163  Ntot 2829085
7164  Ntot 2875334
7165  Ntot 2684058
7166  Ntot 2914473
7167  Ntot 2896728
7168  Ntot 2790747
7169  Ntot 2706259
7170  Ntot 2712238
7171  Ntot 2711553
7172  Ntot 2795459
7173  Ntot 2681402
7174  Ntot 2677581
7175  Ntot 2761633
7176  Ntot 2921562
7177  Ntot 2729493
7178  Ntot 2728339
7179  Ntot 2790869
7180  Ntot 2823072
7181  Ntot 2806976
7182  Ntot 2596516
7183  Ntot 2742957
7184  Ntot 2636087
7185  Ntot 2753710
7186  Ntot 2917107
7187  Ntot 2562775
7188  Ntot 2719448
7189  Ntot 2831482
7190  Ntot 2761063
7191  Ntot 2764035
7192  Ntot 2863936
7193  Ntot 2706664
7194  Ntot 2707056
7195  Ntot 2895615
7196  Ntot 2710746
7197  Ntot 2582269
7198  Ntot 2723383
7199  Ntot 2683285
7200  Ntot 2995100
7201  Ntot 2899098
7202  Ntot 2832512
7203  Ntot 2629677
7204  Ntot 2887648
7205  Ntot 2856033
7206  Ntot 2557258
7207  Ntot 2698661
7208  Ntot 2816315
7209  Ntot 2870829
7210  Ntot 2854616
7211  Ntot 2764624
7212  Ntot 2741504
7213  Ntot 2728880
7214  Ntot 2807765
7215  Ntot 2624062
7216  Ntot 2812343
7217  Ntot 2597951
7218  Ntot 2734375
7219  Ntot 2670353
7220  Ntot 2598997
7221  Ntot 2917316
7222  Ntot 2763368
7223  Ntot 2734221
7224  Ntot 2700220
7225  Ntot 2858599
7226  Ntot 2837645
7227  Ntot 2819636
7228  Ntot 2882756
7229  Ntot 2844648
7230  Ntot 2913164
7231  Ntot 2718133
7232  Ntot 2811934
7233  Ntot 2958444
7234  Ntot 2718305
7235  Ntot 2895653
7236  Ntot 2685573
7237  Ntot 2760544
7238  Ntot 2654660
7239  Ntot 2828988
7240  Ntot 2711618
7241  Ntot 2660466
7242  Ntot 2781631
7243  Ntot 2925829
7244  Ntot 2762209
7245  Ntot 2604017
7246  Ntot 2803476
7247  Ntot 2661148
7248  Ntot 2684423
7249  Ntot 2882144
7250  Ntot 2668376
7251  Ntot 2634158
7252  Ntot 2792521
7253  Ntot 2774713
7254  Ntot 2482857
7255  Ntot 2576355
7256  Ntot 2733049
7257  Ntot 2747486
7258  Ntot 2781195
7259  Ntot 2769944
7260  Ntot 2551964
7261  Ntot 2809897
7262  Ntot 2716573
7263  Ntot 2736255
7264  Ntot 2677388
7265  Ntot 2734993
7266  Ntot 2930070
7267  Ntot 2805345
7268  Ntot 2618310
7269  Ntot 2849051
7270  Ntot 2689484
7271  Ntot 2769461
7272  Ntot 2803017
7273  Ntot 2878339
7274  Ntot 3004933
7275  Ntot 2919986
7276  Ntot 2749423
7277  Ntot 2745513
7278  Ntot 2753593
7279  Ntot 2840568
7280  Ntot 2564689
7281  Ntot 2857304
7282  Ntot 2784504
7283  Ntot 2733817
7284  Ntot 2855286
7285  Ntot 2609923
7286  Ntot 2692624
7287  Ntot 2680956
7288  Ntot 2765075
7289  Ntot 2715422
7290  Ntot 2734896
7291  Ntot 2936609
7292  Ntot 2709351
7293  Ntot 2624967
7294  Ntot 2800023
7295  Ntot 2649164
7296  Ntot 2537597
7297  Ntot 2750126
7298  Ntot 2725025
7299  Ntot 2809556
7300  Ntot 2861352
7301  Ntot 2744596
7302  Ntot 2818377
7303  Ntot 2767898
7304  Ntot 2647164
7305  Ntot 2883551
7306  Ntot 2884986
7307  Ntot 2783107
7308  Ntot 2732771
7309  Ntot 2551996
7310  Ntot 2668902
7311  Ntot 2568280
7312  Ntot 2794189
7313  Ntot 2783763
7314  Ntot 2766383
7315  Ntot 2803608
7316  Ntot 2786747
7317  Ntot 2678232
7318  Ntot 2744303
7319  Ntot 2857315
7320  Ntot 2654030
7321  Ntot 2864744
7322  Ntot 2713450
7323  Ntot 2607739
7324  Ntot 2868785
7325  Ntot 2747504
7326  Ntot 2784983
7327  Ntot 2772594
7328  Ntot 2944399
7329  Ntot 2651215
7330  Ntot 2772946
7331  Ntot 2939941
7332  Ntot 2883065
7333  Ntot 2794397
7334  Ntot 2689832
7335  Ntot 2664746
7336  Ntot 2751397
7337  Ntot 2796013
7338  Ntot 2675815
7339  Ntot 2722584
7340  Ntot 2746141
7341  Ntot 2794092
7342  Ntot 2832315
7343  Ntot 2758694
7344  Ntot 2825987
7345  Ntot 2771458
7346  Ntot 2568276
7347  Ntot 2771710
7348  Ntot 2761499
7349  Ntot 2727653
7350  Ntot 2688454
7351  Ntot 2773471
7352  Ntot 2695023
7353  Ntot 2746523
7354  Ntot 2698405
7355  Ntot 2737498
7356  Ntot 2817714
7357  Ntot 2945128
7358  Ntot 2736902
7359  Ntot 2803598
7360  Ntot 2716216
7361  Ntot 2680854
7362  Ntot 2675966
7363  Ntot 2424636
7364  Ntot 2893234
7365  Ntot 2766560
7366  Ntot 2613541
7367  Ntot 2770408
7368  Ntot 2718949
7369  Ntot 2793492
7370  Ntot 2788100
7371  Ntot 2738961
7372  Ntot 2882792
7373  Ntot 2825183
7374  Ntot 2800694
7375  Ntot 2737300
7376  Ntot 2965525
7377  Ntot 2779474
7378  Ntot 2868592
7379  Ntot 2991815
7380  Ntot 2791511
7381  Ntot 2636787
7382  Ntot 2608546
7383  Ntot 2661912
7384  Ntot 3072045
7385  Ntot 2825989
7386  Ntot 2651868
7387  Ntot 2607582
7388  Ntot 2848827
7389  Ntot 2685651
7390  Ntot 2887629
7391  Ntot 2648779
7392  Ntot 2742430
7393  Ntot 2679949
7394  Ntot 2829641
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7396  Ntot 2746138
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7398  Ntot 2714905
7399  Ntot 2764079
7400  Ntot 2792636
7401  Ntot 2810990
7402  Ntot 2751000
7403  Ntot 2760429
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7405  Ntot 2706960
7406  Ntot 2734062
7407  Ntot 2624179
7408  Ntot 2945262
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7410  Ntot 2594662
7411  Ntot 2656788
7412  Ntot 2718969
7413  Ntot 2710140
7414  Ntot 2747217
7415  Ntot 2678888
7416  Ntot 2743590
7417  Ntot 2771003
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7420  Ntot 2745438
7421  Ntot 2791988
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7423  Ntot 2705234
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7425  Ntot 2727546
7426  Ntot 2783812
7427  Ntot 2695609
7428  Ntot 2644914
7429  Ntot 2715664
7430  Ntot 2816754
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7432  Ntot 2803345
7433  Ntot 2804834
7434  Ntot 2748804
7435  Ntot 2806260
7436  Ntot 2720003
7437  Ntot 2798239
7438  Ntot 2646771
7439  Ntot 2591024
7440  Ntot 2701767
7441  Ntot 2784903
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7443  Ntot 2734946
7444  Ntot 2702902
7445  Ntot 2738299
7446  Ntot 2734197
7447  Ntot 2731526
7448  Ntot 2982280
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7450  Ntot 2810025
7451  Ntot 2811274
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7455  Ntot 2806941
7456  Ntot 2770170
7457  Ntot 2767204
7458  Ntot 2610479
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7460  Ntot 2759707
7461  Ntot 2646836
7462  Ntot 2643139
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7464  Ntot 2750128
7465  Ntot 2876520
7466  Ntot 2762127
7467  Ntot 2888269
7468  Ntot 2868227
7469  Ntot 2629621
7470  Ntot 2911301
7471  Ntot 2807263
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7473  Ntot 2711011
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7475  Ntot 2738537
7476  Ntot 2815079
7477  Ntot 2703062
7478  Ntot 2653816
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7480  Ntot 2814736
7481  Ntot 2559940
7482  Ntot 2845633
7483  Ntot 2751211
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7485  Ntot 2703309
7486  Ntot 2718339
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7488  Ntot 2813541
7489  Ntot 2780425
7490  Ntot 2878554
7491  Ntot 2794301
7492  Ntot 2698533
7493  Ntot 2783261
7494  Ntot 2641797
7495  Ntot 2809511
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7498  Ntot 2809099
7499  Ntot 2685252
7500  Ntot 2730635
7501  Ntot 2715645
7502  Ntot 2588322
7503  Ntot 2765018
7504  Ntot 2764519
7505  Ntot 2717056
7506  Ntot 2980226
7507  Ntot 2808401
7508  Ntot 2795163
7509  Ntot 2793468
7510  Ntot 2746401
7511  Ntot 2664564
7512  Ntot 2535486
7513  Ntot 2645685
7514  Ntot 2929454
7515  Ntot 2706668
7516  Ntot 2782198
7517  Ntot 2874880
7518  Ntot 2812599
7519  Ntot 2664326
7520  Ntot 2698136
7521  Ntot 2822197
7522  Ntot 2607421
7523  Ntot 2786298
7524  Ntot 2898647
7525  Ntot 2946223
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7527  Ntot 2664136
7528  Ntot 2782000
7529  Ntot 2941238
7530  Ntot 2738026
7531  Ntot 2709483
7532  Ntot 2997226
7533  Ntot 2668023
7534  Ntot 2763189
7535  Ntot 2633219
7536  Ntot 2663564
7537  Ntot 3044757
7538  Ntot 2839378
7539  Ntot 2639584
7540  Ntot 2613518
7541  Ntot 2816225
7542  Ntot 2753130
7543  Ntot 2786227
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7545  Ntot 2589564
7546  Ntot 2674419
7547  Ntot 2818689
7548  Ntot 2779525
7549  Ntot 2871104
7550  Ntot 3033351
7551  Ntot 2902731
7552  Ntot 2714005
7553  Ntot 2758477
7554  Ntot 2676732
7555  Ntot 2854340
7556  Ntot 2945891
7557  Ntot 2533676
7558  Ntot 2790877
7559  Ntot 2720958
7560  Ntot 2679913
7561  Ntot 2605522
7562  Ntot 2889211
7563  Ntot 2974280
7564  Ntot 2863648
7565  Ntot 2784487
7566  Ntot 2863384
7567  Ntot 2756719
7568  Ntot 2548458
7569  Ntot 2684009
7570  Ntot 2546513
7571  Ntot 2690837
7572  Ntot 2699131
7573  Ntot 2641286
7574  Ntot 2943418
7575  Ntot 2800299
7576  Ntot 2850207
7577  Ntot 2880634
7578  Ntot 2726818
7579  Ntot 2831551
7580  Ntot 2634635
7581  Ntot 2645213
7582  Ntot 2801620
7583  Ntot 2717115
7584  Ntot 2734190
7585  Ntot 2590227
7586  Ntot 2750498
7587  Ntot 2816149
7588  Ntot 2627830
7589  Ntot 2727767
7590  Ntot 2858464
7591  Ntot 2776058
7592  Ntot 2872610
7593  Ntot 2944542
7594  Ntot 2648773
7595  Ntot 2811437
7596  Ntot 2679154
7597  Ntot 2746138
7598  Ntot 2676668
7599  Ntot 2836047
7600  Ntot 3028606
7601  Ntot 2625986
7602  Ntot 2688915
7603  Ntot 2667662
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7605  Ntot 2699184
7606  Ntot 2728069
7607  Ntot 2748616
7608  Ntot 2757258
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7610  Ntot 2846596
7611  Ntot 2805467
7612  Ntot 2555298
7613  Ntot 2676266
7614  Ntot 2655701
7615  Ntot 2599218
7616  Ntot 2493680
7617  Ntot 2981626
7618  Ntot 2773431
7619  Ntot 2753645
7620  Ntot 2866821
7621  Ntot 2698518
7622  Ntot 2640959
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7624  Ntot 2786290
7625  Ntot 2694855
7626  Ntot 2610436
7627  Ntot 2796692
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7629  Ntot 2989461
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7631  Ntot 2660829
7632  Ntot 2749482
7633  Ntot 2745298
7634  Ntot 2969551
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7636  Ntot 2611897
7637  Ntot 2750302
7638  Ntot 2885774
7639  Ntot 2707851
7640  Ntot 2601263
7641  Ntot 2840895
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7644  Ntot 2774498
7645  Ntot 2872185
7646  Ntot 2821002
7647  Ntot 2704339
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7650  Ntot 2535390
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7652  Ntot 2836876
7653  Ntot 2675028
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7660  Ntot 2909328
7661  Ntot 2829231
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7664  Ntot 2801269
7665  Ntot 2705447
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7667  Ntot 2594474
7668  Ntot 2604416
7669  Ntot 2776743
7670  Ntot 2771840
7671  Ntot 2871087
7672  Ntot 2804791
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7675  Ntot 2725692
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7677  Ntot 2718585
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7680  Ntot 2824574
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7685  Ntot 2841796
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7687  Ntot 2816993
7688  Ntot 2729995
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7698  Ntot 2980521
7699  Ntot 2787016
7700  Ntot 2472291
7701  Ntot 2878361
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7703  Ntot 2640379
7704  Ntot 2664466
7705  Ntot 2613331
7706  Ntot 2772021
7707  Ntot 2922267
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7709  Ntot 2671767
7710  Ntot 2753235
7711  Ntot 2718554
7712  Ntot 2522797
7713  Ntot 2646916
7714  Ntot 2630313
7715  Ntot 2728080
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7718  Ntot 2698059
7719  Ntot 2876833
7720  Ntot 2904238
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7722  Ntot 2823542
7723  Ntot 2716915
7724  Ntot 2778810
7725  Ntot 2600586
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7730  Ntot 2671317
7731  Ntot 2919287
7732  Ntot 2784260
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7734  Ntot 2781380
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7747  Ntot 2713004
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7750  Ntot 2824411
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7752  Ntot 2695694
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7755  Ntot 2879688
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7760  Ntot 2690313
7761  Ntot 2894470
7762  Ntot 2897093
7763  Ntot 2645671
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7767  Ntot 2647913
7768  Ntot 2653006
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7770  Ntot 2639851
7771  Ntot 2949098
7772  Ntot 2824254
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7774  Ntot 2709159
7775  Ntot 2794595
7776  Ntot 2678366
7777  Ntot 2826921
7778  Ntot 2669020
7779  Ntot 2950612
7780  Ntot 2884591
7781  Ntot 2820534
7782  Ntot 2880002
7783  Ntot 2771755
7784  Ntot 2880204
7785  Ntot 2721310
7786  Ntot 2839335
7787  Ntot 2693944
7788  Ntot 2679607
7789  Ntot 2687743
7790  Ntot 2606537
7791  Ntot 2846658
7792  Ntot 2749281
7793  Ntot 2710029
7794  Ntot 2751329
7795  Ntot 2796902
7796  Ntot 2584064
7797  Ntot 2721090
7798  Ntot 2597057
7799  Ntot 2606733
7800  Ntot 2677129
7801  Ntot 2681565
7802  Ntot 2676908
7803  Ntot 2695283
7804  Ntot 2714871
7805  Ntot 2656766
7806  Ntot 2849868
7807  Ntot 2742201
7808  Ntot 2705858
7809  Ntot 2775202
7810  Ntot 2681405
7811  Ntot 2635709
7812  Ntot 2644555
7813  Ntot 2940240
7814  Ntot 2794240
7815  Ntot 2651909
7816  Ntot 2592783
7817  Ntot 2755574
7818  Ntot 2793961
7819  Ntot 2691515
7820  Ntot 2755201
7821  Ntot 2856862
7822  Ntot 2600204
7823  Ntot 2535870
7824  Ntot 2863925
7825  Ntot 2746839
7826  Ntot 2690400
7827  Ntot 2840167
7828  Ntot 2760648
7829  Ntot 2728360
7830  Ntot 2697621
7831  Ntot 2727999
7832  Ntot 3042527
7833  Ntot 2709410
7834  Ntot 2785672
7835  Ntot 2851594
7836  Ntot 2629397
7837  Ntot 2714025
7838  Ntot 2889439
7839  Ntot 2653113
7840  Ntot 2768097
7841  Ntot 2789911
7842  Ntot 2591721
7843  Ntot 2974143
7844  Ntot 2776665
7845  Ntot 2688173
7846  Ntot 2722656
7847  Ntot 2709426
7848  Ntot 2775091
7849  Ntot 2718883
7850  Ntot 2690590
7851  Ntot 2802262
7852  Ntot 2959204
7853  Ntot 2714741
7854  Ntot 2752415
7855  Ntot 2784746
7856  Ntot 2791317
7857  Ntot 2764451
7858  Ntot 2763898
7859  Ntot 2892414
7860  Ntot 2641047
7861  Ntot 2858958
7862  Ntot 2577338
7863  Ntot 2742623
7864  Ntot 2788102
7865  Ntot 2861853
7866  Ntot 2599207
7867  Ntot 2796223
7868  Ntot 2651202
7869  Ntot 2604723
7870  Ntot 2589028
7871  Ntot 2610593
7872  Ntot 2866018
7873  Ntot 2608610
7874  Ntot 2822113
7875  Ntot 2906067
7876  Ntot 2860496
7877  Ntot 2713630
7878  Ntot 2775121
7879  Ntot 2826352
7880  Ntot 2769077
7881  Ntot 2681953
7882  Ntot 2700047
7883  Ntot 2761738
7884  Ntot 2851502
7885  Ntot 2573959
7886  Ntot 2763520
7887  Ntot 2615179
7888  Ntot 2680226
7889  Ntot 2705338
7890  Ntot 2595730
7891  Ntot 2881176
7892  Ntot 2776899
7893  Ntot 2671223
7894  Ntot 2610091
7895  Ntot 2886888
7896  Ntot 2753672
7897  Ntot 2725083
7898  Ntot 2885957
7899  Ntot 2884658
7900  Ntot 2751327
7901  Ntot 2760175
7902  Ntot 2850907
7903  Ntot 2825583
7904  Ntot 2759704
7905  Ntot 2763491
7906  Ntot 2724279
7907  Ntot 2905094
7908  Ntot 2685159
7909  Ntot 2734605
7910  Ntot 2723355
7911  Ntot 2760078
7912  Ntot 2817550
7913  Ntot 2746704
7914  Ntot 2897464
7915  Ntot 2671421
7916  Ntot 2701790
7917  Ntot 2870387
7918  Ntot 2672884
7919  Ntot 2683374
7920  Ntot 2866139
7921  Ntot 2731400
7922  Ntot 2733201
7923  Ntot 2590538
7924  Ntot 2674876
7925  Ntot 2639958
7926  Ntot 2775986
7927  Ntot 2711926
7928  Ntot 2741431
7929  Ntot 2818241
7930  Ntot 2906163
7931  Ntot 2790415
7932  Ntot 2614199
7933  Ntot 2940297
7934  Ntot 2726436
7935  Ntot 2968753
7936  Ntot 2686995
7937  Ntot 2567601
7938  Ntot 2883368
7939  Ntot 2670161
7940  Ntot 2802903
7941  Ntot 2909171
7942  Ntot 2650048
7943  Ntot 2708731
7944  Ntot 2878721
7945  Ntot 2691710
7946  Ntot 2785172
7947  Ntot 2683896
7948  Ntot 2679735
7949  Ntot 2706773
7950  Ntot 2777208
7951  Ntot 2807560
7952  Ntot 2805740
7953  Ntot 2822383
7954  Ntot 2685295
7955  Ntot 2655276
7956  Ntot 3091779
7957  Ntot 2598536
7958  Ntot 2658416
7959  Ntot 2899020
7960  Ntot 2770040
7961  Ntot 2771887
7962  Ntot 2687629
7963  Ntot 2776638
7964  Ntot 2639553
7965  Ntot 2824384
7966  Ntot 2783073
7967  Ntot 2564021
7968  Ntot 2836112
7969  Ntot 2744166
7970  Ntot 2678376
7971  Ntot 2706102
7972  Ntot 2686363
7973  Ntot 2750487
7974  Ntot 2946741
7975  Ntot 2735600
7976  Ntot 2976168
7977  Ntot 2737977
7978  Ntot 2659634
7979  Ntot 2669421
7980  Ntot 2803665
7981  Ntot 2725855
7982  Ntot 2857499
7983  Ntot 2701180
7984  Ntot 2826000
7985  Ntot 3077251
7986  Ntot 2847065
7987  Ntot 2654232
7988  Ntot 2653071
7989  Ntot 2741470
7990  Ntot 2624201
7991  Ntot 2855183
7992  Ntot 2619949
7993  Ntot 2599104
7994  Ntot 2617731
7995  Ntot 2792049
7996  Ntot 2696287
7997  Ntot 2889156
7998  Ntot 2716294
7999  Ntot 2852192
8000  Ntot 2883079
8001  Ntot 2703590
8002  Ntot 2669742
8003  Ntot 2852919
8004  Ntot 2905768
8005  Ntot 2630951
8006  Ntot 2819109
8007  Ntot 2630890
8008  Ntot 2663921
8009  Ntot 2833222
8010  Ntot 2654983
8011  Ntot 2802253
8012  Ntot 2737234
8013  Ntot 2747154
8014  Ntot 2575824
8015  Ntot 2744176
8016  Ntot 2793555
8017  Ntot 2603953
8018  Ntot 2783963
8019  Ntot 2657814
8020  Ntot 2564325
8021  Ntot 2748213
8022  Ntot 2799845
8023  Ntot 2663807
8024  Ntot 2785673
8025  Ntot 2712958
8026  Ntot 2616009
8027  Ntot 2779304
8028  Ntot 2751354
8029  Ntot 2758560
8030  Ntot 2839554
8031  Ntot 2964739
8032  Ntot 2909592
8033  Ntot 2712301
8034  Ntot 2687618
8035  Ntot 3109144
8036  Ntot 2672905
8037  Ntot 2768910
8038  Ntot 2727514
8039  Ntot 2703597
8040  Ntot 2770510
8041  Ntot 2640602
8042  Ntot 2568519
8043  Ntot 2677221
8044  Ntot 2738646
8045  Ntot 2988796
8046  Ntot 2627768
8047  Ntot 2690231
8048  Ntot 2633587
8049  Ntot 2727224
8050  Ntot 2831185
8051  Ntot 2866165
8052  Ntot 2639976
8053  Ntot 2836401
8054  Ntot 2549738
8055  Ntot 2720699
8056  Ntot 2743449
8057  Ntot 2590181
8058  Ntot 2655234
8059  Ntot 2629186
8060  Ntot 2738746
8061  Ntot 2900127
8062  Ntot 2866255
8063  Ntot 2735608
8064  Ntot 2568856
8065  Ntot 2828666
8066  Ntot 2801622
8067  Ntot 2993184
8068  Ntot 2847849
8069  Ntot 2779548
8070  Ntot 2802503
8071  Ntot 2601532
8072  Ntot 2687538
8073  Ntot 2745610
8074  Ntot 2833959
8075  Ntot 2761266
8076  Ntot 2711534
8077  Ntot 2884652
8078  Ntot 2751975
8079  Ntot 2738490
8080  Ntot 2516154
8081  Ntot 2542888
8082  Ntot 2940705
8083  Ntot 2657184
8084  Ntot 2718083
8085  Ntot 2590089
8086  Ntot 2683078
8087  Ntot 2820787
8088  Ntot 2701995
8089  Ntot 2677249
8090  Ntot 2656719
8091  Ntot 2696389
8092  Ntot 2708612
8093  Ntot 2626979
8094  Ntot 2934879
8095  Ntot 2713435
8096  Ntot 2792651
8097  Ntot 2880498
8098  Ntot 2662570
8099  Ntot 2644382
8100  Ntot 2760527
8101  Ntot 2893573
8102  Ntot 2766446
8103  Ntot 2706543
8104  Ntot 2639893
8105  Ntot 2841527
8106  Ntot 2859831
8107  Ntot 2823105
8108  Ntot 2841704
8109  Ntot 2783095
8110  Ntot 2830450
8111  Ntot 2613998
8112  Ntot 2607728
8113  Ntot 2672924
8114  Ntot 2747938
8115  Ntot 2771430
8116  Ntot 2635464
8117  Ntot 2741962
8118  Ntot 2830649
8119  Ntot 2626848
8120  Ntot 2635537
8121  Ntot 2789817
8122  Ntot 2641467
8123  Ntot 2710603
8124  Ntot 2664500
8125  Ntot 2619146
8126  Ntot 2723827
8127  Ntot 2798186
8128  Ntot 2764589
8129  Ntot 2685216
8130  Ntot 2602367
8131  Ntot 2766106
8132  Ntot 2532576
8133  Ntot 2724649
8134  Ntot 2702369
8135  Ntot 2660407
8136  Ntot 2803117
8137  Ntot 2921833
8138  Ntot 2752600
8139  Ntot 2987187
8140  Ntot 2644351
8141  Ntot 2692372
8142  Ntot 2794889
8143  Ntot 2875603
8144  Ntot 2674530
8145  Ntot 2856008
8146  Ntot 2702475
8147  Ntot 2879903
8148  Ntot 2774259
8149  Ntot 2773298
8150  Ntot 2778822
8151  Ntot 2810533
8152  Ntot 2756251
8153  Ntot 2685845
8154  Ntot 2593942
8155  Ntot 2882588
8156  Ntot 2727934
8157  Ntot 2853072
8158  Ntot 2855755
8159  Ntot 2984912
8160  Ntot 2932099
8161  Ntot 2813241
8162  Ntot 3018820
8163  Ntot 2610616
8164  Ntot 2775098
8165  Ntot 2641757
8166  Ntot 2740643
8167  Ntot 2848361
8168  Ntot 2816325
8169  Ntot 2627502
8170  Ntot 2654387
8171  Ntot 2844628
8172  Ntot 2902511
8173  Ntot 2907330
8174  Ntot 2814370
8175  Ntot 2911758
8176  Ntot 2571854
8177  Ntot 2657546
8178  Ntot 3002021
8179  Ntot 2773011
8180  Ntot 2552494
8181  Ntot 2688862
8182  Ntot 2768716
8183  Ntot 2763314
8184  Ntot 2774513
8185  Ntot 2559734
8186  Ntot 2784233
8187  Ntot 2560192
8188  Ntot 2689225
8189  Ntot 2678328
8190  Ntot 2710275
8191  Ntot 2728327
8192  Ntot 2779472
8193  Ntot 2920350
8194  Ntot 2675604
8195  Ntot 2592527
8196  Ntot 2930392
8197  Ntot 2702298
8198  Ntot 2707171
8199  Ntot 2667949
8200  Ntot 2750905
8201  Ntot 2622257
8202  Ntot 2774548
8203  Ntot 2743099
8204  Ntot 2689499
8205  Ntot 2718885
8206  Ntot 2764887
8207  Ntot 2621997
8208  Ntot 2797698
8209  Ntot 2938054
8210  Ntot 2725041
8211  Ntot 2916386
8212  Ntot 2889871
8213  Ntot 2849550
8214  Ntot 2776411
8215  Ntot 2662486
8216  Ntot 2777806
8217  Ntot 2693126
8218  Ntot 2923521
8219  Ntot 2732894
8220  Ntot 2662347
8221  Ntot 2732478
8222  Ntot 2823077
8223  Ntot 2822760
8224  Ntot 2827993
8225  Ntot 2849302
8226  Ntot 2841404
8227  Ntot 2733322
8228  Ntot 2825543
8229  Ntot 2858916
8230  Ntot 2546373
8231  Ntot 2573857
8232  Ntot 2801862
8233  Ntot 2680781
8234  Ntot 2752127
8235  Ntot 2707849
8236  Ntot 2887782
8237  Ntot 2722365
8238  Ntot 2636783
8239  Ntot 2570388
8240  Ntot 2686797
8241  Ntot 2619332
8242  Ntot 2812651
8243  Ntot 2720234
8244  Ntot 2667139
8245  Ntot 2832349
8246  Ntot 2813141
8247  Ntot 2851963
8248  Ntot 2655792
8249  Ntot 2797952
8250  Ntot 2835807
8251  Ntot 2641520
8252  Ntot 2423307
8253  Ntot 2711281
8254  Ntot 2675673
8255  Ntot 2765043
8256  Ntot 2678185
8257  Ntot 2885162
8258  Ntot 2675632
8259  Ntot 2820930
8260  Ntot 2882567
8261  Ntot 2703316
8262  Ntot 2732459
8263  Ntot 2778049
8264  Ntot 2621348
8265  Ntot 2714248
8266  Ntot 2658253
8267  Ntot 2909456
8268  Ntot 2820441
8269  Ntot 2950685
8270  Ntot 2687754
8271  Ntot 2744942
8272  Ntot 2636123
8273  Ntot 2897031
8274  Ntot 2921359
8275  Ntot 2597311
8276  Ntot 2818791
8277  Ntot 2874017
8278  Ntot 2810254
8279  Ntot 2815749
8280  Ntot 2667726
8281  Ntot 2635690
8282  Ntot 2610914
8283  Ntot 2768197
8284  Ntot 2567013
8285  Ntot 2903379
8286  Ntot 2759929
8287  Ntot 2748991
8288  Ntot 2700154
8289  Ntot 2641697
8290  Ntot 2741526
8291  Ntot 2881871
8292  Ntot 3061542
8293  Ntot 2631411
8294  Ntot 2946168
8295  Ntot 2602847
8296  Ntot 2714198
8297  Ntot 2735536
8298  Ntot 2712307
8299  Ntot 2935655
8300  Ntot 2647723
8301  Ntot 2875094
8302  Ntot 2739256
8303  Ntot 2773564
8304  Ntot 2859511
8305  Ntot 2644092
8306  Ntot 2658433
8307  Ntot 2799985
8308  Ntot 2828418
8309  Ntot 2747547
8310  Ntot 2710107
8311  Ntot 3027766
8312  Ntot 2871720
8313  Ntot 2825758
8314  Ntot 2825043
8315  Ntot 2607915
8316  Ntot 2884332
8317  Ntot 2737413
8318  Ntot 2793429
8319  Ntot 2818970
8320  Ntot 2564231
8321  Ntot 2542732
8322  Ntot 2839772
8323  Ntot 2889426
8324  Ntot 2858402
8325  Ntot 2817091
8326  Ntot 2693876
8327  Ntot 2685112
8328  Ntot 2615655
8329  Ntot 2783163
8330  Ntot 2693141
8331  Ntot 2784592
8332  Ntot 2751369
8333  Ntot 2723055
8334  Ntot 2708871
8335  Ntot 3000252
8336  Ntot 2687746
8337  Ntot 2590006
8338  Ntot 2752563
8339  Ntot 2803726
8340  Ntot 2654327
8341  Ntot 2681721
8342  Ntot 2922665
8343  Ntot 2713743
8344  Ntot 2655935
8345  Ntot 2796283
8346  Ntot 2803825
8347  Ntot 2767997
8348  Ntot 2621248
8349  Ntot 2680474
8350  Ntot 2653970
8351  Ntot 2735517
8352  Ntot 2798946
8353  Ntot 2782035
8354  Ntot 2755242
8355  Ntot 2810693
8356  Ntot 2896095
8357  Ntot 2725741
8358  Ntot 2756051
8359  Ntot 2660451
8360  Ntot 2662369
8361  Ntot 2774145
8362  Ntot 2615712
8363  Ntot 2726833
8364  Ntot 2682450
8365  Ntot 2814418
8366  Ntot 2726376
8367  Ntot 2775813
8368  Ntot 2870030
8369  Ntot 2898060
8370  Ntot 2813987
8371  Ntot 2698065
8372  Ntot 2852155
8373  Ntot 2741862
8374  Ntot 2738701
8375  Ntot 2720110
8376  Ntot 2786788
8377  Ntot 2709817
8378  Ntot 2949288
8379  Ntot 2689723
8380  Ntot 2790208
8381  Ntot 2917381
8382  Ntot 2793692
8383  Ntot 2731195
8384  Ntot 2822803
8385  Ntot 2813775
8386  Ntot 2634208
8387  Ntot 2713128
8388  Ntot 2671255
8389  Ntot 2688707
8390  Ntot 2582157
8391  Ntot 2707581
8392  Ntot 2713349
8393  Ntot 2613689
8394  Ntot 2920865
8395  Ntot 2726150
8396  Ntot 2923515
8397  Ntot 2765174
8398  Ntot 2781414
8399  Ntot 2837605
8400  Ntot 2762395
8401  Ntot 2815390
8402  Ntot 2803429
8403  Ntot 2834548
8404  Ntot 2714532
8405  Ntot 2821580
8406  Ntot 2768135
8407  Ntot 2909839
8408  Ntot 2851957
8409  Ntot 2665311
8410  Ntot 2989650
8411  Ntot 2781772
8412  Ntot 2892431
8413  Ntot 2643450
8414  Ntot 2834293
8415  Ntot 2558227
8416  Ntot 2696189
8417  Ntot 2733188
8418  Ntot 2756604
8419  Ntot 2803764
8420  Ntot 2778791
8421  Ntot 2814220
8422  Ntot 2686102
8423  Ntot 2596172
8424  Ntot 2766271
8425  Ntot 2728775
8426  Ntot 2701394
8427  Ntot 2837552
8428  Ntot 2799129
8429  Ntot 2757840
8430  Ntot 2806463
8431  Ntot 2643091
8432  Ntot 2804908
8433  Ntot 2653950
8434  Ntot 2650335
8435  Ntot 2767472
8436  Ntot 2765029
8437  Ntot 2574722
8438  Ntot 2929835
8439  Ntot 2771313
8440  Ntot 2820598
8441  Ntot 2518578
8442  Ntot 2851584
8443  Ntot 2476677
8444  Ntot 2801688
8445  Ntot 2808183
8446  Ntot 2718560
8447  Ntot 2766674
8448  Ntot 2623059
8449  Ntot 2734529
8450  Ntot 2727793
8451  Ntot 2952467
8452  Ntot 2615791
8453  Ntot 2777403
8454  Ntot 2653860
8455  Ntot 2654975
8456  Ntot 2881962
8457  Ntot 2975920
8458  Ntot 2813198
8459  Ntot 2822255
8460  Ntot 2681979
8461  Ntot 2606821
8462  Ntot 2797067
8463  Ntot 2840522
8464  Ntot 2713150
8465  Ntot 3030826
8466  Ntot 2598215
8467  Ntot 2832752
8468  Ntot 2822955
8469  Ntot 2611412
8470  Ntot 2744329
8471  Ntot 2724866
8472  Ntot 2648615
8473  Ntot 2747994
8474  Ntot 2801944
8475  Ntot 2733589
8476  Ntot 2601824
8477  Ntot 2740497
8478  Ntot 2866211
8479  Ntot 2927678
8480  Ntot 2766774
8481  Ntot 2694087
8482  Ntot 2630239
8483  Ntot 2894266
8484  Ntot 2679731
8485  Ntot 2696603
8486  Ntot 2608137
8487  Ntot 2730342
8488  Ntot 2626061
8489  Ntot 2685657
8490  Ntot 2719307
8491  Ntot 3069966
8492  Ntot 2799600
8493  Ntot 2717289
8494  Ntot 2847880
8495  Ntot 2635754
8496  Ntot 2729026
8497  Ntot 2921378
8498  Ntot 2835129
8499  Ntot 2815944
8500  Ntot 2706231
8501  Ntot 2658240
8502  Ntot 2733162
8503  Ntot 2836018
8504  Ntot 2644378
8505  Ntot 2839054
8506  Ntot 2749099
8507  Ntot 2768302
8508  Ntot 2576435
8509  Ntot 2811993
8510  Ntot 2844319
8511  Ntot 2870404
8512  Ntot 2597429
8513  Ntot 2760981
8514  Ntot 2696406
8515  Ntot 2744693
8516  Ntot 2899257
8517  Ntot 2976386
8518  Ntot 2721377
8519  Ntot 2685725
8520  Ntot 2781652
8521  Ntot 2602658
8522  Ntot 2707029
8523  Ntot 2763303
8524  Ntot 2744798
8525  Ntot 2711436
8526  Ntot 2608319
8527  Ntot 2574180
8528  Ntot 2660491
8529  Ntot 2934915
8530  Ntot 2692001
8531  Ntot 2733591
8532  Ntot 2984073
8533  Ntot 2652542
8534  Ntot 2771948
8535  Ntot 2801539
8536  Ntot 2758732
8537  Ntot 2826465
8538  Ntot 2757509
8539  Ntot 2793805
8540  Ntot 2677873
8541  Ntot 2775341
8542  Ntot 2742152
8543  Ntot 2922959
8544  Ntot 2822098
8545  Ntot 2754449
8546  Ntot 2850422
8547  Ntot 2786090
8548  Ntot 2735960
8549  Ntot 2656499
8550  Ntot 2664258
8551  Ntot 2839767
8552  Ntot 2810769
8553  Ntot 2756658
8554  Ntot 2612348
8555  Ntot 2611172
8556  Ntot 2812557
8557  Ntot 2742816
8558  Ntot 2877862
8559  Ntot 2789010
8560  Ntot 2547577
8561  Ntot 2734831
8562  Ntot 2865214
8563  Ntot 2862508
8564  Ntot 2725060
8565  Ntot 2644223
8566  Ntot 2687441
8567  Ntot 2769892
8568  Ntot 2750064
8569  Ntot 2883958
8570  Ntot 2787500
8571  Ntot 2745289
8572  Ntot 2721716
8573  Ntot 2683088
8574  Ntot 2842107
8575  Ntot 2784626
8576  Ntot 2609140
8577  Ntot 2815739
8578  Ntot 2726591
8579  Ntot 2965136
8580  Ntot 2865017
8581  Ntot 2823079
8582  Ntot 2680970
8583  Ntot 3131036
8584  Ntot 2722881
8585  Ntot 2747309
8586  Ntot 2697730
8587  Ntot 2595978
8588  Ntot 2733287
8589  Ntot 2682194
8590  Ntot 2772789
8591  Ntot 2852234
8592  Ntot 2614139
8593  Ntot 2771178
8594  Ntot 2735857
8595  Ntot 2615872
8596  Ntot 2665455
8597  Ntot 2865919
8598  Ntot 2950088
8599  Ntot 2967217
8600  Ntot 2765191
8601  Ntot 2744039
8602  Ntot 2864822
8603  Ntot 2962553
8604  Ntot 2607926
8605  Ntot 2725139
8606  Ntot 2745005
8607  Ntot 2748101
8608  Ntot 2656623
8609  Ntot 2980841
8610  Ntot 2766099
8611  Ntot 2710571
8612  Ntot 2716461
8613  Ntot 2645566
8614  Ntot 2799454
8615  Ntot 2749838
8616  Ntot 2742112
8617  Ntot 2736733
8618  Ntot 2771251
8619  Ntot 2587397
8620  Ntot 2803709
8621  Ntot 2687465
8622  Ntot 2815484
8623  Ntot 2934659
8624  Ntot 2825214
8625  Ntot 2542434
8626  Ntot 2842618
8627  Ntot 2654338
8628  Ntot 2784397
8629  Ntot 2663236
8630  Ntot 2796728
8631  Ntot 2670100
8632  Ntot 2980506
8633  Ntot 2833215
8634  Ntot 2620781
8635  Ntot 2482985
8636  Ntot 2756481
8637  Ntot 2801345
8638  Ntot 2782802
8639  Ntot 2640947
8640  Ntot 2852923
8641  Ntot 2612147
8642  Ntot 2861973
8643  Ntot 2698427
8644  Ntot 2463228
8645  Ntot 2859533
8646  Ntot 2681648
8647  Ntot 2798477
8648  Ntot 2833845
8649  Ntot 2786038
8650  Ntot 2647180
8651  Ntot 2759054
8652  Ntot 2843401
8653  Ntot 2767493
8654  Ntot 2693108
8655  Ntot 2753285
8656  Ntot 2671113
8657  Ntot 2679834
8658  Ntot 2716261
8659  Ntot 2853430
8660  Ntot 2814347
8661  Ntot 2734087
8662  Ntot 2714199
8663  Ntot 2837573
8664  Ntot 2804119
8665  Ntot 2717295
8666  Ntot 3046727
8667  Ntot 2713174
8668  Ntot 2809755
8669  Ntot 2817385
8670  Ntot 2643297
8671  Ntot 2752623
8672  Ntot 2876975
8673  Ntot 2921357
8674  Ntot 2679000
8675  Ntot 2584617
8676  Ntot 2801203
8677  Ntot 2785183
8678  Ntot 2707974
8679  Ntot 2796487
8680  Ntot 2663882
8681  Ntot 2699038
8682  Ntot 2646630
8683  Ntot 2624751
8684  Ntot 2799492
8685  Ntot 2733241
8686  Ntot 2789664
8687  Ntot 2702267
8688  Ntot 2665852
8689  Ntot 2910733
8690  Ntot 2855713
8691  Ntot 2745825
8692  Ntot 2607787
8693  Ntot 2730626
8694  Ntot 2746890
8695  Ntot 2774867
8696  Ntot 2667135
8697  Ntot 2898353
8698  Ntot 2630390
8699  Ntot 2716989
8700  Ntot 2699516
8701  Ntot 2695038
8702  Ntot 2661853
8703  Ntot 2727681
8704  Ntot 2662029
8705  Ntot 3003499
8706  Ntot 2715477
8707  Ntot 2733960
8708  Ntot 2957501
8709  Ntot 2818533
8710  Ntot 2836551
8711  Ntot 2839715
8712  Ntot 2778427
8713  Ntot 2893540
8714  Ntot 2766485
8715  Ntot 2679509
8716  Ntot 2635003
8717  Ntot 2551108
8718  Ntot 2645935
8719  Ntot 2737508
8720  Ntot 2714354
8721  Ntot 2740703
8722  Ntot 2755142
8723  Ntot 2727892
8724  Ntot 2650460
8725  Ntot 2754784
8726  Ntot 2699951
8727  Ntot 2867419
8728  Ntot 2773762
8729  Ntot 2911283
8730  Ntot 2718499
8731  Ntot 2768684
8732  Ntot 2831576
8733  Ntot 2772215
8734  Ntot 2652046
8735  Ntot 2734715
8736  Ntot 2642751
8737  Ntot 2848682
8738  Ntot 2665395
8739  Ntot 2736383
8740  Ntot 2873638
8741  Ntot 2756324
8742  Ntot 2686972
8743  Ntot 2685925
8744  Ntot 2797201
8745  Ntot 2686667
8746  Ntot 2679896
8747  Ntot 2650033
8748  Ntot 2931947
8749  Ntot 2649647
8750  Ntot 2752379
8751  Ntot 2821613
8752  Ntot 2799285
8753  Ntot 2632553
8754  Ntot 2747229
8755  Ntot 2429807
8756  Ntot 2550374
8757  Ntot 2757318
8758  Ntot 2651297
8759  Ntot 2819455
8760  Ntot 2728774
8761  Ntot 2832070
8762  Ntot 2904950
8763  Ntot 2991826
8764  Ntot 2696181
8765  Ntot 2856188
8766  Ntot 2716794
8767  Ntot 2621251
8768  Ntot 2782282
8769  Ntot 2833056
8770  Ntot 2776219
8771  Ntot 2685789
8772  Ntot 2702789
8773  Ntot 2812244
8774  Ntot 2784137
8775  Ntot 2687990
8776  Ntot 2639066
8777  Ntot 2649630
8778  Ntot 2698677
8779  Ntot 2844120
8780  Ntot 2894771
8781  Ntot 2765993
8782  Ntot 2734103
8783  Ntot 2666597
8784  Ntot 2695070
8785  Ntot 2673909
8786  Ntot 2830671
8787  Ntot 2744950
8788  Ntot 2904794
8789  Ntot 2768001
8790  Ntot 2790189
8791  Ntot 2929298
8792  Ntot 2881503
8793  Ntot 2656495
8794  Ntot 2787287
8795  Ntot 2845501
8796  Ntot 2860484
8797  Ntot 2958865
8798  Ntot 2762767
8799  Ntot 2751693
8800  Ntot 2780558
8801  Ntot 2731111
8802  Ntot 2775967
8803  Ntot 2590591
8804  Ntot 2639094
8805  Ntot 2705806
8806  Ntot 2759962
8807  Ntot 2892272
8808  Ntot 2625374
8809  Ntot 2829397
8810  Ntot 2637064
8811  Ntot 2543195
8812  Ntot 2808163
8813  Ntot 2588287
8814  Ntot 2645625
8815  Ntot 2789133
8816  Ntot 2678584
8817  Ntot 2692412
8818  Ntot 2822584
8819  Ntot 2893780
8820  Ntot 2632503
8821  Ntot 2495306
8822  Ntot 2722038
8823  Ntot 2879168
8824  Ntot 2607179
8825  Ntot 2632335
8826  Ntot 2747787
8827  Ntot 2817750
8828  Ntot 2657674
8829  Ntot 2864821
8830  Ntot 2619611
8831  Ntot 2753991
8832  Ntot 2768037
8833  Ntot 2587353
8834  Ntot 2877495
8835  Ntot 2621636
8836  Ntot 2850568
8837  Ntot 2780324
8838  Ntot 2651613
8839  Ntot 2770037
8840  Ntot 2656153
8841  Ntot 2685203
8842  Ntot 2727007
8843  Ntot 2795299
8844  Ntot 2818786
8845  Ntot 2612645
8846  Ntot 2820899
8847  Ntot 2558672
8848  Ntot 2836723
8849  Ntot 2720090
8850  Ntot 2931028
8851  Ntot 3017796
8852  Ntot 2628260
8853  Ntot 2645144
8854  Ntot 2999001
8855  Ntot 2841635
8856  Ntot 2880898
8857  Ntot 2692359
8858  Ntot 2719036
8859  Ntot 2691623
8860  Ntot 2634287
8861  Ntot 2749105
8862  Ntot 2604056
8863  Ntot 3004215
8864  Ntot 2712141
8865  Ntot 2704027
8866  Ntot 2777941
8867  Ntot 2600785
8868  Ntot 2613226
8869  Ntot 2738554
8870  Ntot 2769980
8871  Ntot 2768765
8872  Ntot 2920228
8873  Ntot 2814626
8874  Ntot 2789586
8875  Ntot 2936403
8876  Ntot 2910587
8877  Ntot 2625902
8878  Ntot 2746020
8879  Ntot 2760826
8880  Ntot 2582103
8881  Ntot 2778263
8882  Ntot 2804347
8883  Ntot 2582210
8884  Ntot 2778222
8885  Ntot 2764038
8886  Ntot 2634437
8887  Ntot 2697708
8888  Ntot 2727046
8889  Ntot 2802289
8890  Ntot 2888366
8891  Ntot 2670083
8892  Ntot 2694514
8893  Ntot 2492074
8894  Ntot 2681834
8895  Ntot 2647175
8896  Ntot 2719491
8897  Ntot 2882997
8898  Ntot 2754149
8899  Ntot 2691908
8900  Ntot 2676061
8901  Ntot 2928848
8902  Ntot 2643772
8903  Ntot 2908095
8904  Ntot 2973430
8905  Ntot 2876686
8906  Ntot 2698343
8907  Ntot 2884218
8908  Ntot 2762543
8909  Ntot 2554516
8910  Ntot 2784224
8911  Ntot 2779147
8912  Ntot 2895691
8913  Ntot 2741531
8914  Ntot 2713366
8915  Ntot 2930166
8916  Ntot 2818846
8917  Ntot 2687750
8918  Ntot 2823870
8919  Ntot 2616388
8920  Ntot 2742432
8921  Ntot 2822811
8922  Ntot 2908159
8923  Ntot 2722973
8924  Ntot 2804698
8925  Ntot 2795622
8926  Ntot 2876252
8927  Ntot 2803852
8928  Ntot 2740400
8929  Ntot 2681481
8930  Ntot 2793107
8931  Ntot 2722318
8932  Ntot 2740691
8933  Ntot 2907390
8934  Ntot 2840759
8935  Ntot 2676859
8936  Ntot 2766791
8937  Ntot 2756786
8938  Ntot 2850107
8939  Ntot 2834693
8940  Ntot 2625883
8941  Ntot 2836081
8942  Ntot 2835595
8943  Ntot 2714574
8944  Ntot 2823495
8945  Ntot 2886136
8946  Ntot 2692340
8947  Ntot 2518480
8948  Ntot 2739453
8949  Ntot 2588136
8950  Ntot 2585670
8951  Ntot 2758267
8952  Ntot 2775918
8953  Ntot 2900047
8954  Ntot 2783319
8955  Ntot 2715052
8956  Ntot 2803056
8957  Ntot 2688606
8958  Ntot 2667906
8959  Ntot 2785073
8960  Ntot 2850399
8961  Ntot 2838911
8962  Ntot 2754021
8963  Ntot 2947698
8964  Ntot 2753617
8965  Ntot 2874831
8966  Ntot 2808031
8967  Ntot 2848566
8968  Ntot 2732696
8969  Ntot 2777746
8970  Ntot 2825770
8971  Ntot 2588977
8972  Ntot 2965866
8973  Ntot 2722070
8974  Ntot 2896534
8975  Ntot 2820996
8976  Ntot 2989523
8977  Ntot 2924795
8978  Ntot 2675958
8979  Ntot 2680273
8980  Ntot 2721626
8981  Ntot 2663762
8982  Ntot 2718770
8983  Ntot 2765222
8984  Ntot 2626512
8985  Ntot 2778872
8986  Ntot 2697135
8987  Ntot 2875051
8988  Ntot 2816867
8989  Ntot 2796492
8990  Ntot 2781104
8991  Ntot 2781772
8992  Ntot 2716802
8993  Ntot 2668657
8994  Ntot 2854058
8995  Ntot 2693126
8996  Ntot 2842669
8997  Ntot 2727690
8998  Ntot 2856604
8999  Ntot 2720551
9000  Ntot 2770917
9001  Ntot 2781310
9002  Ntot 2775807
9003  Ntot 2768401
9004  Ntot 2566152
9005  Ntot 2825473
9006  Ntot 2893691
9007  Ntot 2741783
9008  Ntot 2738812
9009  Ntot 2503327
9010  Ntot 2953664
9011  Ntot 2616323
9012  Ntot 2849891
9013  Ntot 2807061
9014  Ntot 2784870
9015  Ntot 2761996
9016  Ntot 2934845
9017  Ntot 2925962
9018  Ntot 2822294
9019  Ntot 2857953
9020  Ntot 2791675
9021  Ntot 2825880
9022  Ntot 2940333
9023  Ntot 2571541
9024  Ntot 2822669
9025  Ntot 2850174
9026  Ntot 2712949
9027  Ntot 2899091
9028  Ntot 2653915
9029  Ntot 2654359
9030  Ntot 2760607
9031  Ntot 2782988
9032  Ntot 2685567
9033  Ntot 2718544
9034  Ntot 2672820
9035  Ntot 2904154
9036  Ntot 2784747
9037  Ntot 2758848
9038  Ntot 2699350
9039  Ntot 2711237
9040  Ntot 2790510
9041  Ntot 2718522
9042  Ntot 2876295
9043  Ntot 2765735
9044  Ntot 2894518
9045  Ntot 2731476
9046  Ntot 2799439
9047  Ntot 2888418
9048  Ntot 2726980
9049  Ntot 2859137
9050  Ntot 2744776
9051  Ntot 2827483
9052  Ntot 2704203
9053  Ntot 2516548
9054  Ntot 2850618
9055  Ntot 2536769
9056  Ntot 2880929
9057  Ntot 2930158
9058  Ntot 2741557
9059  Ntot 2789700
9060  Ntot 2763861
9061  Ntot 2937617
9062  Ntot 2868382
9063  Ntot 2723631
9064  Ntot 2701304
9065  Ntot 2675927
9066  Ntot 2617183
9067  Ntot 2646329
9068  Ntot 2788114
9069  Ntot 2710630
9070  Ntot 2906545
9071  Ntot 2768463
9072  Ntot 2649601
9073  Ntot 2747428
9074  Ntot 2605844
9075  Ntot 2708823
9076  Ntot 2758684
9077  Ntot 2828664
9078  Ntot 2840152
9079  Ntot 2653288
9080  Ntot 2840556
9081  Ntot 2769199
9082  Ntot 2751006
9083  Ntot 2716637
9084  Ntot 2666403
9085  Ntot 2782981
9086  Ntot 2653150
9087  Ntot 2694272
9088  Ntot 2846564
9089  Ntot 2702188
9090  Ntot 2646894
9091  Ntot 2831728
9092  Ntot 2813753
9093  Ntot 2647651
9094  Ntot 2755441
9095  Ntot 2718339
9096  Ntot 2791912
9097  Ntot 2933078
9098  Ntot 2737350
9099  Ntot 2759554
9100  Ntot 2784275
9101  Ntot 2711499
9102  Ntot 2764520
9103  Ntot 2570372
9104  Ntot 2762871
9105  Ntot 2691309
9106  Ntot 2928660
9107  Ntot 2653959
9108  Ntot 2805704
9109  Ntot 2698211
9110  Ntot 2855500
9111  Ntot 2746780
9112  Ntot 2672171
9113  Ntot 2602533
9114  Ntot 2714394
9115  Ntot 2521984
9116  Ntot 2750569
9117  Ntot 2640154
9118  Ntot 2624763
9119  Ntot 2610262
9120  Ntot 2680646
9121  Ntot 2713315
9122  Ntot 2918862
9123  Ntot 2870609
9124  Ntot 2665867
9125  Ntot 2955638
9126  Ntot 2647420
9127  Ntot 2884238
9128  Ntot 2576078
9129  Ntot 2764548
9130  Ntot 2595162
9131  Ntot 2642716
9132  Ntot 2726765
9133  Ntot 2864179
9134  Ntot 2713262
9135  Ntot 2587817
9136  Ntot 2846346
9137  Ntot 2653721
9138  Ntot 2802160
9139  Ntot 2680614
9140  Ntot 2797370
9141  Ntot 2671235
9142  Ntot 2778616
9143  Ntot 2768483
9144  Ntot 2730811
9145  Ntot 2813918
9146  Ntot 2726374
9147  Ntot 2717464
9148  Ntot 2778737
9149  Ntot 2835651
9150  Ntot 2829153
9151  Ntot 2673174
9152  Ntot 2676920
9153  Ntot 2849396
9154  Ntot 2604062
9155  Ntot 2783812
9156  Ntot 2831490
9157  Ntot 2816296
9158  Ntot 2747239
9159  Ntot 2902754
9160  Ntot 2762179
9161  Ntot 2749151
9162  Ntot 2678842
9163  Ntot 2784604
9164  Ntot 2917067
9165  Ntot 2688422
9166  Ntot 2698310
9167  Ntot 2807382
9168  Ntot 2992505
9169  Ntot 2745185
9170  Ntot 2706137
9171  Ntot 2744579
9172  Ntot 2745713
9173  Ntot 2718296
9174  Ntot 2799885
9175  Ntot 2483223
9176  Ntot 2795741
9177  Ntot 2750201
9178  Ntot 2605068
9179  Ntot 2876186
9180  Ntot 2715093
9181  Ntot 2953466
9182  Ntot 2784106
9183  Ntot 2679266
9184  Ntot 2886985
9185  Ntot 2808985
9186  Ntot 2714393
9187  Ntot 2847932
9188  Ntot 2732983
9189  Ntot 2602898
9190  Ntot 2878733
9191  Ntot 2652177
9192  Ntot 2603623
9193  Ntot 2764210
9194  Ntot 2741210
9195  Ntot 2631693
9196  Ntot 2672724
9197  Ntot 2699820
9198  Ntot 2726469
9199  Ntot 2757036
9200  Ntot 2819631
9201  Ntot 2878581
9202  Ntot 2785717
9203  Ntot 2614610
9204  Ntot 2646650
9205  Ntot 2833472
9206  Ntot 2759788
9207  Ntot 2719902
9208  Ntot 2609622
9209  Ntot 2963154
9210  Ntot 2606876
9211  Ntot 2796935
9212  Ntot 2991702
9213  Ntot 2743345
9214  Ntot 3009854
9215  Ntot 2634377
9216  Ntot 2582389
9217  Ntot 2780688
9218  Ntot 2837557
9219  Ntot 2658234
9220  Ntot 2781445
9221  Ntot 2782042
9222  Ntot 2938741
9223  Ntot 2705477
9224  Ntot 2843912
9225  Ntot 2688821
9226  Ntot 2879704
9227  Ntot 2816043
9228  Ntot 2595232
9229  Ntot 2758055
9230  Ntot 2775830
9231  Ntot 2761377
9232  Ntot 2873870
9233  Ntot 2614633
9234  Ntot 2564354
9235  Ntot 2679963
9236  Ntot 2741553
9237  Ntot 2795307
9238  Ntot 2698042
9239  Ntot 2614421
9240  Ntot 2669946
9241  Ntot 3048902
9242  Ntot 2743986
9243  Ntot 2722091
9244  Ntot 2738908
9245  Ntot 2650988
9246  Ntot 2765699
9247  Ntot 2769277
9248  Ntot 2761191
9249  Ntot 2713934
9250  Ntot 2861678
9251  Ntot 2649025
9252  Ntot 2677641
9253  Ntot 2636276
9254  Ntot 2830224
9255  Ntot 2707906
9256  Ntot 2781518
9257  Ntot 2780165
9258  Ntot 2782914
9259  Ntot 2810313
9260  Ntot 2875854
9261  Ntot 2720577
9262  Ntot 2680965
9263  Ntot 2570048
9264  Ntot 2806832
9265  Ntot 2818835
9266  Ntot 2714808
9267  Ntot 2681059
9268  Ntot 2838444
9269  Ntot 2630366
9270  Ntot 2783913
9271  Ntot 2858610
9272  Ntot 2933508
9273  Ntot 3064672
9274  Ntot 2808859
9275  Ntot 2762209
9276  Ntot 2769850
9277  Ntot 2853062
9278  Ntot 2797613
9279  Ntot 2667890
9280  Ntot 2832592
9281  Ntot 2594293
9282  Ntot 2598615
9283  Ntot 2918830
9284  Ntot 2887050
9285  Ntot 2538723
9286  Ntot 2763546
9287  Ntot 2769336
9288  Ntot 2894294
9289  Ntot 2769161
9290  Ntot 2587770
9291  Ntot 2685258
9292  Ntot 2684202
9293  Ntot 2788151
9294  Ntot 2735721
9295  Ntot 2746560
9296  Ntot 2783183
9297  Ntot 2728551
9298  Ntot 2804458
9299  Ntot 2630328
9300  Ntot 2616923
9301  Ntot 2803049
9302  Ntot 2646740
9303  Ntot 2531722
9304  Ntot 2640199
9305  Ntot 2715514
9306  Ntot 2817314
9307  Ntot 2944152
9308  Ntot 2597850
9309  Ntot 2953799
9310  Ntot 2765681
9311  Ntot 3108999
9312  Ntot 2671838
9313  Ntot 2873132
9314  Ntot 2633515
9315  Ntot 2770069
9316  Ntot 2489797
9317  Ntot 2683907
9318  Ntot 2612362
9319  Ntot 2701212
9320  Ntot 2838646
9321  Ntot 2808024
9322  Ntot 2905626
9323  Ntot 2638795
9324  Ntot 2797415
9325  Ntot 2687145
9326  Ntot 2634246
9327  Ntot 2554806
9328  Ntot 2754690
9329  Ntot 2680907
9330  Ntot 2764304
9331  Ntot 2748248
9332  Ntot 2826319
9333  Ntot 2747429
9334  Ntot 2801837
9335  Ntot 2648616
9336  Ntot 2631048
9337  Ntot 2695564
9338  Ntot 2822140
9339  Ntot 2685967
9340  Ntot 2706969
9341  Ntot 2694132
9342  Ntot 2585485
9343  Ntot 2707851
9344  Ntot 2624399
9345  Ntot 2769091
9346  Ntot 2644744
9347  Ntot 2611212
9348  Ntot 2894930
9349  Ntot 2815605
9350  Ntot 2928259
9351  Ntot 2589767
9352  Ntot 2862338
9353  Ntot 2805422
9354  Ntot 2597335
9355  Ntot 2833036
9356  Ntot 2835147
9357  Ntot 2695386
9358  Ntot 2804395
9359  Ntot 2865374
9360  Ntot 2926767
9361  Ntot 2827927
9362  Ntot 2667505
9363  Ntot 2769514
9364  Ntot 2837254
9365  Ntot 2696172
9366  Ntot 2570140
9367  Ntot 2699183
9368  Ntot 2746574
9369  Ntot 2720180
9370  Ntot 2824429
9371  Ntot 2618059
9372  Ntot 2682025
9373  Ntot 2886254
9374  Ntot 2739914
9375  Ntot 2793271
9376  Ntot 2762860
9377  Ntot 2691113
9378  Ntot 2710564
9379  Ntot 2725164
9380  Ntot 2643107
9381  Ntot 2609126
9382  Ntot 2604028
9383  Ntot 2675765
9384  Ntot 2693859
9385  Ntot 2722787
9386  Ntot 2906374
9387  Ntot 2617418
9388  Ntot 2724711
9389  Ntot 2662474
9390  Ntot 2734133
9391  Ntot 2882539
9392  Ntot 2618096
9393  Ntot 2751548
9394  Ntot 2829569
9395  Ntot 2664440
9396  Ntot 2757182
9397  Ntot 2716704
9398  Ntot 2601154
9399  Ntot 2797531
9400  Ntot 2729098
9401  Ntot 2808988
9402  Ntot 2643068
9403  Ntot 2749474
9404  Ntot 2906958
9405  Ntot 2833538
9406  Ntot 2792519
9407  Ntot 2833622
9408  Ntot 2800710
9409  Ntot 2557734
9410  Ntot 2729315
9411  Ntot 2725579
9412  Ntot 2680333
9413  Ntot 2762039
9414  Ntot 2793059
9415  Ntot 2780048
9416  Ntot 2781621
9417  Ntot 2891446
9418  Ntot 2719335
9419  Ntot 2594987
9420  Ntot 2653993
9421  Ntot 2672034
9422  Ntot 2868192
9423  Ntot 2809867
9424  Ntot 3015090
9425  Ntot 2932278
9426  Ntot 2664357
9427  Ntot 2627515
9428  Ntot 2728256
9429  Ntot 2543909
9430  Ntot 2683028
9431  Ntot 2941141
9432  Ntot 2626462
9433  Ntot 2554529
9434  Ntot 2905414
9435  Ntot 2860671
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9437  Ntot 2552841
9438  Ntot 2783595
9439  Ntot 2674635
9440  Ntot 2816662
9441  Ntot 2737250
9442  Ntot 2715715
9443  Ntot 2615842
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9445  Ntot 2830366
9446  Ntot 2662202
9447  Ntot 2687001
9448  Ntot 2598313
9449  Ntot 2789485
9450  Ntot 2877721
9451  Ntot 2900990
9452  Ntot 2796377
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9454  Ntot 2853073
9455  Ntot 2629892
9456  Ntot 2918313
9457  Ntot 2751760
9458  Ntot 2732562
9459  Ntot 2724052
9460  Ntot 2708313
9461  Ntot 2703903
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9464  Ntot 2655985
9465  Ntot 2704935
9466  Ntot 2673682
9467  Ntot 2748756
9468  Ntot 2732214
9469  Ntot 2793457
9470  Ntot 2938210
9471  Ntot 2695305
9472  Ntot 2738422
9473  Ntot 2630255
9474  Ntot 2593671
9475  Ntot 2813999
9476  Ntot 2861424
9477  Ntot 2572497
9478  Ntot 2637827
9479  Ntot 2740150
9480  Ntot 2563782
9481  Ntot 2839111
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9483  Ntot 2791141
9484  Ntot 2823700
9485  Ntot 2692832
9486  Ntot 2777475
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9488  Ntot 2664977
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9490  Ntot 2709669
9491  Ntot 3059543
9492  Ntot 2857347
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9494  Ntot 2698986
9495  Ntot 2696256
9496  Ntot 2758874
9497  Ntot 2557092
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9499  Ntot 2754746
9500  Ntot 2896745
9501  Ntot 2828882
9502  Ntot 2879246
9503  Ntot 2685796
9504  Ntot 2688448
9505  Ntot 2763419
9506  Ntot 2780193
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9508  Ntot 2803950
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9510  Ntot 2830559
9511  Ntot 2764564
9512  Ntot 2786702
9513  Ntot 2699069
9514  Ntot 2722004
9515  Ntot 2920000
9516  Ntot 2681579
9517  Ntot 2722379
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9519  Ntot 2749458
9520  Ntot 2720171
9521  Ntot 2763797
9522  Ntot 2635061
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9524  Ntot 2722188
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9526  Ntot 2715869
9527  Ntot 2703952
9528  Ntot 2653647
9529  Ntot 2800626
9530  Ntot 2742453
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9532  Ntot 2693670
9533  Ntot 2843854
9534  Ntot 2757599
9535  Ntot 2660949
9536  Ntot 2745777
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9538  Ntot 2867392
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9540  Ntot 2754184
9541  Ntot 2868505
9542  Ntot 2610950
9543  Ntot 2725713
9544  Ntot 2609610
9545  Ntot 2839011
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9548  Ntot 2914325
9549  Ntot 2715668
9550  Ntot 2710623
9551  Ntot 3007109
9552  Ntot 2750309
9553  Ntot 2753436
9554  Ntot 2722522
9555  Ntot 2859330
9556  Ntot 2693577
9557  Ntot 2868832
9558  Ntot 2727400
9559  Ntot 2729063
9560  Ntot 2544246
9561  Ntot 2879956
9562  Ntot 3020509
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9564  Ntot 2602994
9565  Ntot 2738636
9566  Ntot 2771576
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9568  Ntot 2599835
9569  Ntot 2839934
9570  Ntot 2701272
9571  Ntot 2645761
9572  Ntot 2800838
9573  Ntot 2733913
9574  Ntot 2704487
9575  Ntot 2777214
9576  Ntot 2692959
9577  Ntot 2609366
9578  Ntot 2896898
9579  Ntot 2692714
9580  Ntot 2784725
9581  Ntot 2689341
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9583  Ntot 2595071
9584  Ntot 2920401
9585  Ntot 2753127
9586  Ntot 2612793
9587  Ntot 2748771
9588  Ntot 2705341
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9590  Ntot 2827026
9591  Ntot 2852728
9592  Ntot 2780789
9593  Ntot 2734838
9594  Ntot 2721173
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9596  Ntot 2626144
9597  Ntot 2705460
9598  Ntot 2819224
9599  Ntot 2616602
9600  Ntot 2720204
9601  Ntot 2740781
9602  Ntot 2749377
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9604  Ntot 2656747
9605  Ntot 2692661
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9607  Ntot 2961221
9608  Ntot 2812309
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9610  Ntot 2707309
9611  Ntot 2716140
9612  Ntot 2987708
9613  Ntot 2788279
9614  Ntot 2782240
9615  Ntot 2903495
9616  Ntot 2574323
9617  Ntot 2732915
9618  Ntot 2723254
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9620  Ntot 2759651
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9622  Ntot 2825947
9623  Ntot 2733627
9624  Ntot 2919994
9625  Ntot 2699566
9626  Ntot 2845509
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9628  Ntot 2624340
9629  Ntot 2732808
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9631  Ntot 2631365
9632  Ntot 2778288
9633  Ntot 2584859
9634  Ntot 2633782
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9636  Ntot 2690421
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9638  Ntot 2608306
9639  Ntot 2874998
9640  Ntot 2751886
9641  Ntot 2829898
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9644  Ntot 2898353
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9650  Ntot 2673820
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9656  Ntot 2687876
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9660  Ntot 2726668
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9664  Ntot 2600724
9665  Ntot 2681338
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9667  Ntot 2747893
9668  Ntot 2714678
9669  Ntot 2597454
9670  Ntot 2742830
9671  Ntot 2722163
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9674  Ntot 2675033
9675  Ntot 2605683
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9680  Ntot 2751169
9681  Ntot 2721879
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9685  Ntot 2839604
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9688  Ntot 2729982
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9690  Ntot 2573568
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9699  Ntot 2749518
9700  Ntot 2676270
9701  Ntot 2657016
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9705  Ntot 2969636
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9710  Ntot 2752908
9711  Ntot 2822264
9712  Ntot 2876522
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9714  Ntot 2992533
9715  Ntot 2654105
9716  Ntot 2821846
9717  Ntot 2702639
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9720  Ntot 2797980
9721  Ntot 2940664
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9725  Ntot 2620705
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9727  Ntot 2804245
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9730  Ntot 2809117
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9732  Ntot 2981185
9733  Ntot 2749011
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9735  Ntot 2551125
9736  Ntot 2899457
9737  Ntot 2791903
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9740  Ntot 2639256
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9747  Ntot 2948225
9748  Ntot 2558167
9749  Ntot 2673717
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9760  Ntot 2782849
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9764  Ntot 2603102
9765  Ntot 2749810
9766  Ntot 2813873
9767  Ntot 2573610
9768  Ntot 2909845
9769  Ntot 2737430
9770  Ntot 2746274
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9775  Ntot 2964225
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9780  Ntot 2640420
9781  Ntot 2681837
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9790  Ntot 2852133
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9795  Ntot 2756587
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9799  Ntot 2694416
9800  Ntot 2743312
9801  Ntot 2703467
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9810  Ntot 2689979
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9813  Ntot 2798866
9814  Ntot 2634814
9815  Ntot 2613319
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9823  Ntot 2828132
9824  Ntot 2653651
9825  Ntot 2781903
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9827  Ntot 2564246
9828  Ntot 2853987
9829  Ntot 2756132
9830  Ntot 2629103
9831  Ntot 2774777
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9834  Ntot 2734798
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9836  Ntot 2914636
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9839  Ntot 2789772
9840  Ntot 2628156
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9847  Ntot 2686239
9848  Ntot 2911590
9849  Ntot 2607560
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9860  Ntot 2667310
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9864  Ntot 2786909
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9999  Ntot 2816404
10000 Ntot 2845986
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12000 Ntot 2948012

$layers
$layers[[1]]
mapping: xintercept = ~quants 
geom_vline: na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 

$layers[[2]]
geom_density: na.rm = FALSE, orientation = NA, outline.type = upper
stat_density: na.rm = FALSE, orientation = NA
position_identity 

$layers[[3]]
mapping: label = ~paste("Posterior mean: ", post.mean, "\nPosterior sd  : ", post.sd, sep = ""), vjust = 1, hjust = 1, x = Inf, y = Inf 
geom_text: parse = FALSE, check_overlap = FALSE, size.unit = mm, na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 


$scales
<ggproto object: Class ScalesList, gg>
    add: function
    add_defaults: function
    add_missing: function
    backtransform_df: function
    clone: function
    find: function
    get_scales: function
    has_scale: function
    input: function
    map_df: function
    n: function
    non_position_scales: function
    scales: list
    set_palettes: function
    train_df: function
    transform_df: function
    super:  <ggproto object: Class ScalesList, gg>

$guides
<Guides[0] ggproto object>

<empty>

$mapping
$x
<quosure>
expr: ^sample
env:  0x7fc5b0c691c0

$y
<quosure>
expr: ^..density..
env:  0x7fc5b0c691c0

attr(,"class")
[1] "uneval"

$theme
list()

$coordinates
<ggproto object: Class CoordCartesian, Coord, gg>
    aspect: function
    backtransform_range: function
    clip: on
    default: TRUE
    distance: function
    draw_panel: function
    expand: TRUE
    is_free: function
    is_linear: function
    labels: function
    limits: list
    modify_scales: function
    range: function
    render_axis_h: function
    render_axis_v: function
    render_bg: function
    render_fg: function
    reverse: none
    setup_data: function
    setup_layout: function
    setup_panel_guides: function
    setup_panel_params: function
    setup_params: function
    train_panel_guides: function
    transform: function
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$labels$x
[1] "Value of parameter"

$labels$title
[1] "Posterior plots with 95% credible intervals"

$labels$y
[1] "density"

$labels$xintercept
[1] "quants"

$labels$label
[1] "paste(\"Posterior mean: \", post.mean, \"\\nPosterior sd  : \", post.sd, ..."

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attr(,"class")
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A plot of the \(logit(P)\) (the logit of the estimated probability of capture) is shown in ?@fig-btspas-diag1-logitP

$data
                mean         sd      2.5%       25%       50%       75%
logitP[1]  -3.231698 0.13166827 -3.497633 -3.318289 -3.229264 -3.141008
logitP[2]  -2.130373 0.08796830 -2.303911 -2.188610 -2.131085 -2.072361
logitP[3]  -2.548131 0.23811194 -3.018273 -2.707732 -2.542864 -2.382758
logitP[4]  -1.969197 0.08310100 -2.133044 -2.025039 -1.969582 -1.913775
logitP[5]  -3.045225 0.11452159 -3.280643 -3.120658 -3.043189 -2.968012
logitP[6]  -2.984068 0.14661543 -3.285655 -3.080481 -2.980426 -2.883478
logitP[7]  -2.987145 0.17455309 -3.325219 -3.107043 -2.983866 -2.866972
logitP[8]  -3.245073 0.05891636 -3.364442 -3.284550 -3.245334 -3.205374
logitP[9]  -2.902177 0.05657181 -3.014050 -2.939995 -2.901456 -2.864548
logitP[10] -3.168038 0.08219533 -3.330144 -3.223521 -3.167514 -3.112814
logitP[11] -2.684641 0.05859884 -2.802081 -2.723134 -2.684892 -2.645085
logitP[12] -2.074265 0.05413162 -2.182144 -2.110587 -2.074178 -2.038139
logitP[13] -2.179161 0.09858649 -2.374061 -2.244765 -2.179621 -2.111672
logitP[14] -2.116590 0.16351244 -2.440340 -2.225951 -2.112243 -2.004697
logitP[15] -2.147474 0.28629293 -2.716532 -2.335621 -2.143557 -1.955828
logitP[16] -2.750555 0.42629097 -3.595089 -3.025926 -2.733124 -2.467695
               97.5%     Rhat n.eff time.index time       lcl       ucl
logitP[1]  -2.984970 1.000761  6000          1    4 -3.497633 -2.984970
logitP[2]  -1.960295 1.001229  4200          2    5 -2.303911 -1.960295
logitP[3]  -2.107626 1.001593  2400          3    6 -3.018273 -2.107626
logitP[4]  -1.807444 1.000789  6000          4    7 -2.133044 -1.807444
logitP[5]  -2.823893 1.000856  6000          5    8 -3.280643 -2.823893
logitP[6]  -2.707897 1.000810  6000          6    9 -3.285655 -2.707897
logitP[7]  -2.650606 1.000854  6000          7   10 -3.325219 -2.650606
logitP[8]  -3.133552 1.000792  6000          8   11 -3.364442 -3.133552
logitP[9]  -2.792491 1.001348  3300          9   12 -3.014050 -2.792491
logitP[10] -3.007150 1.001269  3800         10   13 -3.330144 -3.007150
logitP[11] -2.569459 1.001283  3700         11   14 -2.802081 -2.569459
logitP[12] -1.967395 1.001042  6000         12   15 -2.182144 -1.967395
logitP[13] -1.988892 1.000818  6000         13   16 -2.374061 -1.988892
logitP[14] -1.803157 1.001025  6000         14   17 -2.440340 -1.803157
logitP[15] -1.593993 1.001405  3000         15   18 -2.716532 -1.593993
logitP[16] -1.937867 1.000932  6000         16   19 -3.595089 -1.937867

$layers
$layers[[1]]
geom_point: na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 

$layers[[2]]
geom_line: na.rm = FALSE, orientation = NA
stat_identity: na.rm = FALSE
position_identity 

$layers[[3]]
mapping: ymin = ~lcl, ymax = ~ucl 
geom_errorbar: na.rm = FALSE, orientation = NA, width = 0.1
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$layers[[4]]
mapping: x = ~x, y = ~y 
geom_point: na.rm = FALSE
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$layers[[5]]
mapping: yintercept = ~yintercept 
geom_hline: na.rm = FALSE
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mapping: yintercept = ~yintercept 
geom_hline: na.rm = FALSE
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$x
<quosure>
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env:  0x7fc5767cfe98

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$labels
$labels$y
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$labels$x
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$labels$title
[1] " \nPlot of logit(p[i]) with 95% credible intervals"

$labels$ymin
[1] "lcl"

$labels$ymax
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$labels$yintercept
[1] "yintercept"


attr(,"class")
[1] "gg"     "ggplot"

A summary of the posterior for each parameter is also available. The estimates of total abundance can be extracted and summarized in a similar fashion as in the other models:

BTSPAS.diag.est1 <-  Petersen::LP_BTSPAS_est (BTSPAS.diag.fit1)
BTSPAS.diag.est1$summary

  N_hat_f N_hat_rn   N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 2750764 104131.5             0.95         Posterior
  N_hat_LCL N_hat_UCL p_model name_model cond.ll n.parms   nobs
1   2557856   2965873      ~1      p: ~1      NA      NA 154081

Or, all of the parameters can be extracted directly from the BTSPAS object as shown below for the total abundance and the total unmarked abundance.

BTSPAS.diag.fit1$fit$summary[ 
    row.names(BTSPAS.diag.fit1$fit$summary) %in% c("Ntot","Utot"),]

        mean       sd    2.5%     25%     50%     75%   97.5%     Rhat n.eff
Ntot 2750764 104131.5 2557856 2679591 2748473 2817332 2965873 1.000899  6000
Utot 2715779 104131.5 2522871 2644606 2713488 2782347 2930888 1.000899  6000

This also includes the Rubin-Brooks-Gelman statistic (\(Rhat\)) on mixing of the chains and the effective sample size of the posterior (after accounting for autocorrelation).

The estimated total abundance from BTSPAS is 2,750,764 (SD 104,132 ) fish.

Samples from the posterior are also included in the sims.matrix, sims.array and sims.list elements of the BTSPAS results object.

It is always important to do model assessment before accepting the results from the model fit. Please check the vignettes of BTSPAS on details on how to interpret the goodness of fit, trace, and autocorrelation plots.

8.3.4.3 Dealing with problems in the data

The BTSPAS package is quite flexible when dealing with problems in the data. Please consult the vignettes with the BTSPAS package for many examples.

8.3.4.3.1 Missing data from some strata

In some cases, data is missing for some temporal strata. For example, river flows could be too high to use the fishwheels safely; crews are sick; data are lost; etc. In these cases, the spline is used to interpolate the run abundance over the missing data.

For example, we will delete data from temporal strata 5 and 8. The resulting summary data is:

xtabs( freq ~ fw1 + fw2, 
       data=data_btspas_diag3.aug[ data_btspas_diag3.aug$fw1 %in% temporal_strata[1:10] &
                                   data_btspas_diag3.aug$fw2 %in% temporal_strata[1:10]  ,],
       exclude=NULL, na.action=na.pass)

    fw2
fw1      0     4     6     7     9    10    11    12    13    14
  0      0 14587  1027  1945  1323   933 57549 14846  5291  9249
  4   1414    51     0     0     0     0     0     0     0     0
  6    180     0    17     0     0     0     0     0     0     0
  7   1167     0     0   168     0     0     0     0     0     0
  9    880     0     0     0    43     0     0     0     0     0
  10   582     0     0     0     0    28     0     0     0     0
  11  7672     0     0     0     0     0   297     0     0     0
  12  5695     0     0     0     0     0     0   313     0     0
  13  3619     0     0     0     0     0     0     0   151     0
  14  4545     0     0     0     0     0     0     0     0   309

Notice that there is no data for temporal strata 5 or 8.

We can also get the summary statistics (\(n_1\), \(m_2\), and \(u_2\)) for this data (and notice the warning messages):

# compute n, m,u
nmu <- Petersen::cap_hist_to_n_m_u(data_btspas_diag3)

Warning in Petersen::cap_hist_to_n_m_u(data_btspas_diag3): *** Caution... Missing value for n1 set to 0 ***

Warning in Petersen::cap_hist_to_n_m_u(data_btspas_diag3): *** Caution... Missing value for u2. These are not set to zero ***

temp <- data.frame(time=nmu$..ts, n1=nmu$n1, m2=nmu$m2, u2=nmu$u2)
temp

   time   n1  m2    u2
1     4 1465  51 14587
2     5    0   0    NA
3     6  197  17  1027
4     7 1335 168  1945
5     8    0   0    NA
6     9  923  43  1323
7    10  610  28   933
8    11 7969 297 57549
9    12 6008 313 14846
10   13 3770 151  5291
11   14 4854 309  9249
12   15 3350 376  4615
13   16 1062 110  1433
14   17  346  40   446
15   18   88  12   120
16   19   20   1    23

Notice that in stratum 5 and 8, the number of releases (\(n_1\)) and recaptures (\(m_2\)) are set to 0 (no data present), but the \(u_2\) values are set to NA (missing) indicating that the number of unmarked fish recaptured is unknown because the trap was not running.

We fit the BTSPAS model in the same way as previously.

BTSPAS.diag.fit3 <- Petersen::LP_BTSPAS_fit_Diag(
         data_btspas_diag3,
         p_model=~1,
         jump.after=10,
         InitialSeed=234234
)

which gives the model fit summary:

The fitted spline curve to the number of unmarked fish available in each recovery sample is shown in ?@fig-btspas-diag3-fit-plot

$data
   time     logUi      logU   logUlcl   logUucl    spline
1     4 12.927686 12.870748 12.628092 13.130223 12.141153
2     5        NA 11.099866  9.621431 12.729523 11.094507
3     6  9.338304  9.589931  9.170411 10.054614 10.479818
4     7  9.641816  9.671664  9.524568  9.822385 10.140268
5     8        NA  9.938060  8.546256 11.524391  9.935922
6     9 10.234017 10.196858  9.925155 10.488059  9.794381
7    10  9.888913  9.874493  9.552240 10.240214  9.660129
8    11 14.246881 14.242148 14.128670 14.351760 14.188454
9    12 12.557340 12.564021 12.458354 12.674500 13.028688
10   13 11.785432 11.788974 11.635265 11.950952 12.056103
11   14 11.883777 11.878763 11.773516 11.983797 11.190367
12   15 10.622351 10.629986 10.530459 10.729449 10.351151
13   16  9.528483  9.555053  9.375053  9.741904  9.461895
14   17  8.241189  8.335322  8.045083  8.636142  8.461110
15   18  6.730651  7.051943  6.568192  7.574096  7.291075
16   19  5.575949  5.881865  5.049610  6.769213  5.894073

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mapping: y = ~logU 
geom_line: na.rm = FALSE, orientation = NA
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mapping: ymin = ~logUlcl, ymax = ~logUucl 
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mapping: y = ~spline 
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[1] "Time Index\nOpen/closed circles - initial and final estimates"

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[1] ""

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[1] "Fitted spline curve with 95% credible intervals for estimated log(U[i])"

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attr(,"class")
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Notice that the curve was interpolated at temporal strata 5 and 8 (with a much larger uncertainty) compared to the other temporal strata.

A plot of the \(logit(P)\) (the logit of the estimated probability of capture) is shown in ?@fig-btspas-diag3-logitP

$data
                mean         sd      2.5%       25%       50%       75%
logitP[1]  -3.244538 0.13257796 -3.513328 -3.333153 -3.242738 -3.152992
logitP[2]  -2.645604 0.57862657 -3.835580 -2.994285 -2.641751 -2.288546
logitP[3]  -2.578597 0.23985347 -3.069219 -2.735522 -2.573735 -2.416246
logitP[4]  -1.966909 0.08287820 -2.129034 -2.022486 -1.966976 -1.910156
logitP[5]  -2.628422 0.56780920 -3.752556 -2.984687 -2.623984 -2.273001
logitP[6]  -2.960155 0.14656117 -3.256647 -3.055440 -2.959173 -2.859740
logitP[7]  -2.986749 0.17994917 -3.358333 -3.102912 -2.980153 -2.862603
logitP[8]  -3.243406 0.05945823 -3.358075 -3.284021 -3.243618 -3.203281
logitP[9]  -2.905233 0.05765564 -3.020753 -2.943317 -2.904222 -2.865531
logitP[10] -3.173755 0.08196937 -3.338911 -3.228708 -3.172940 -3.117523
logitP[11] -2.680429 0.05724164 -2.792317 -2.718878 -2.680332 -2.641420
logitP[12] -2.074375 0.05474580 -2.181102 -2.110295 -2.073864 -2.038187
logitP[13] -2.180072 0.10023464 -2.376577 -2.246388 -2.179985 -2.112735
logitP[14] -2.120235 0.16157727 -2.441884 -2.230467 -2.118523 -2.009281
logitP[15] -2.151095 0.27732567 -2.702129 -2.338798 -2.147739 -1.961460
logitP[16] -2.705442 0.43357932 -3.594034 -2.984441 -2.699150 -2.422858
               97.5%     Rhat n.eff time.index time       lcl       ucl
logitP[1]  -2.991323 1.001036  6000          1    4 -3.513328 -2.991323
logitP[2]  -1.494495 1.001406  3000          2    5 -3.835580 -1.494495
logitP[3]  -2.124042 1.000839  6000          3    6 -3.069219 -2.124042
logitP[4]  -1.807890 1.000794  6000          4    7 -2.129034 -1.807890
logitP[5]  -1.482596 1.000824  6000          5    8 -3.752556 -1.482596
logitP[6]  -2.681456 1.001288  3700          6    9 -3.256647 -2.681456
logitP[7]  -2.651714 1.001782  1900          7   10 -3.358333 -2.651714
logitP[8]  -3.126269 1.000797  6000          8   11 -3.358075 -3.126269
logitP[9]  -2.794718 1.000869  6000          9   12 -3.020753 -2.794718
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In those cases with no data, the capture-probability is estimated from the hierarchical model (at the mean of the model) with high uncertainty, but this not used to estimate \(U_2\) because \(u_2\) is missing.

A summary of the posterior for each parameter is also available. The estimates of total abundance can be extracted and summarized in a similar fashion as in the other models. The estimate is similar to the case where there was no missing data, but the uncertainty is larger.

BTSPAS.diag.est3 <-  Petersen::LP_BTSPAS_est (BTSPAS.diag.fit3)
BTSPAS.diag.est3$summary

  N_hat_f N_hat_rn   N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 2782845 295438.5             0.95         Posterior
  N_hat_LCL N_hat_UCL p_model name_model cond.ll n.parms   nobs
1   2544770   3107300      ~1      p: ~1      NA      NA 145384

8.3.4.3.2 Fixing p’s

In some cases, the second trap is not running and so there are no recaptures of tagged fish and no captures of untagged fish. This usually ends up with 0’s for the total number of untagged fish captured in a temporal stratum, even though fish were released.

We need to set the p’s in these strata to 0 rather than letting BTSPAS impute a value based on the hierarchical model for the p’s.

This is done by passing the temporal stratum where the \(logit(p)\) should be fixed typically at \(logit(p)=-10\) which corresponds to \(p=\) 0.0000454.

For example, to fix the \(logit(p)\) for temporal stratum 12, the following code would be used:

BTSPAS.diag.fit2 <- Petersen::LP_BTSPAS_fit_Diag(
         data_btspas_diag1,
         p_model=~1,
         jump.after=10,
         logitP.fixed=12, logitP.fixed.values=-10, 
         InitialSeed=23943242
)

8.3.4.3.3 Using covariates to model the p’s

BTSPAS also allows you to model the p’s with additional covariates, such a temperature, stream flow, etc. It is not possible to use covariates to model the total number of unmarked fish. A separate data frame must be created with the covariate value for each of the temporal strata at the second wheel.

Here is the data frame with the covariate data. There must be a line for every temporal stratum between the first and last stratum, even those where the traps were not running.

   ..ts2  logflow
2      4 6.564691
3      5 7.077220
4      6 6.884975
5      7 6.984033
6      8 7.923348
7      9 8.072650
8     10 7.817110
9     11 7.681441
10    12 7.607920
11    13 7.438067
12    14 6.391612
13    15 6.276237
14    16 6.464555
15    17 7.159880
16    18 6.559354
17    19 6.113682

A preliminary plot of the empirical logit
shows an approximate quadratic fit to \(log(flow)\), but the uncertainty in each week is enormous! (Figure 21)

Figure 21: Logit(p) vs log(flow)

The summary statistics are:

We can also get the summary statistics (\(n_1\), \(m_2\), and \(u_2\)) for this data:

   time   n1  m2    u2
1     4 1465  51 14587
2     5 1335 146  2854
3     6  197  17  1027
4     7 1335 168  1945
5     8 1653  72  2855
6     9  923  43  1323
7    10  610  28   933
8    11 7969 297 57549
9    12 6008 313 14846
10   13 3770 151  5291
11   14 4854 309  9249
12   15 3350 376  4615
13   16 1062 110  1433
14   17  346  40   446
15   18   88  12   120
16   19   20   1    23

There is no missing data.

We fit the model by modifying the formula for p_model and including the covariate data

BTSPAS.diag.fit4 <- Petersen::LP_BTSPAS_fit_Diag(
         data_btspas_diag1,
         p_model=~logflow+I(logflow^2), p_model_cov=p_cov,
         jump.after=10,
         InitialSeed=23943242
)

Then the estimates can be extracted in the usual fashion:

BTSPAS.diag.est4 <-  Petersen::LP_BTSPAS_est (BTSPAS.diag.fit4)
BTSPAS.diag.est4$summary

  N_hat_f N_hat_rn   N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 2743295 103613.3             0.95         Posterior
  N_hat_LCL N_hat_UCL                 p_model                 name_model
1   2549144   2957416 ~logflow + I(logflow^2) p: ~logflow + I(logflow^2)
  cond.ll n.parms   nobs
1      NA      NA 154081

There is only a minor change in the estimate compared to not using a covariate (seen previously)

8.3.5 Non-diagonal case

In this case, the released fish are not recaptured in a single future temporal stratum, and recaptures takes place for a number of future strata. Here is an example of data collected under this protocol. Fish were tagged and released on the Conne River each day (denoted by the julian day of the year). The fish move upstream and are recaptured at an upstream trap (along with untagged fish).

data(data_btspas_nondiag1)
data_btspas_nondiag1[ grepl("^027",     data_btspas_nondiag1$cap_hist) |
                      grepl("000..027", data_btspas_nondiag1$cap_hist),]

   cap_hist freq
5  027..000  224
20 000..027 1420
28 027..028    1
34 027..029   39
40 027..030   84
47 027..031   35
54 027..032    1
61 027..033    3
68 027..034    1

In temporal stratum 27, 388 fish were released at the first fishwheel, of which 1 were recaptured in the next temporal stratum at the second fishwheel, 39 were recaptured in the following temporal stratum at the second fishwheel etc, and 224 were never seen again. An additional 1420 unmarked fish were captured at the second fishwheel.

Recaptures of marked fish no longer take place along the “diagonal” that was seen in the previous example (only the first 10 strata are shown below):

    fw2
fw1      0    23    24    25    26    27    28    29    30    31    32    33
  0      0     0     0     0     0  1420 27886  9284  4357 11871 14896  9910
  23    21     0     0     0     0     0    11     2     0     0     0     0
  24   132     0     0     0     0     3   118    18     6     3     0     0
  25   328     0     0     0     0     1   149    76    33    16     3     1
  26   294     0     0     0     0     0    42   163    69    34     1     0
  27   224     0     0     0     0     0     1    39    84    35     1     3
  28   100     0     0     0     0     0     0     1    29    60    14     5
  29   266     0     0     0     0     0     0     0     3    82    84    19
  30   291     0     0     0     0     0     0     0     0    20   154    56
  31   191     0     0     0     0     0     0     0     0     0    16    44

The first row (corresponding to fw1=0) are the number of unmarked fish recovered at the second fish wheel in the temporal strata. The first column (corresponding to fw2=0) are the number of released marked fish that were never seen again.

Most fish seem to take between 4 to 10 weeks to move between fish wheels.

Note that the second fishwheel was not operating in temporal strata 23 to 26, so no recoveries are possible, and the number of unmarked fish recaptured is also 0. The capture probability will have to be set to 0 for these temporal strata.

The summary statistics are:

# compute n, m,u
nmu <- Petersen::cap_hist_to_n_m_u(data_btspas_nondiag1)
temp <- data.frame(time =nmu$..ts, 
                   n1   =c(nmu$n1,   rep(NA, length(nmu$u2)-length(nmu$n1))), 
                   m2   =rbind(nmu$m2, matrix(NA, nrow=length(nmu$u2)-length(nmu$n1), ncol=ncol(nmu$m2))) ,          
                   u2   =nmu$u2)
temp

   time  n1 m2.1 m2.2 m2.3 m2.4 m2.5 m2.6 m2.7 m2.8 m2.9 m2.10    u2
1    23  34    0    0    0    0    0   11    2    0    0     0     0
2    24 280    0    0    0    3  118   18    6    3    0     0     0
3    25 607    0    0    1  149   76   33   16    3    1     0     0
4    26 603    0    0   42  163   69   34    1    0    0     0     0
5    27 388    0    1   39   84   35    1    3    1    0     0  1420
6    28 212    0    1   29   60   14    5    2    0    1     0 27886
7    29 468    0    3   82   84   19   11    2    1    0     0  9284
8    30 586    0   20  154   56   53    9    1    1    1     0  4357
9    31 512    0   16   44  146   83   25    3    3    0     1 11871
10   32 458    0    2   52  125   82   13   14    3    0     0 14896
11   33 479    0    0   83  133   43   12    5    2    0     0  9910
12   34 329    0   12   96   60   29    8    0    1    0     0 16526
13   35 248    0   19   72   37   17    7    0    0    0     0 17443
14   36 201    0    7   55   23    6    1    0    0    0     0 16485
15   37 104    0    7    3    3    0    0    0    0    0     0  6776
16   38  NA   NA   NA   NA   NA   NA   NA   NA   NA   NA    NA  4644
17   39  NA   NA   NA   NA   NA   NA   NA   NA   NA   NA    NA  2190
18   40  NA   NA   NA   NA   NA   NA   NA   NA   NA   NA    NA  1066
19   41  NA   NA   NA   NA   NA   NA   NA   NA   NA   NA    NA   166

8.3.5.1 Preliminary screening of the data

A pooled-Petersen estimator would add all of the marked, recaptured and unmarked fish to give an estimate of 283,200 (SE 3,616) fish but can the estimate be trusted?

Let us first examine a plot of the estimated capture efficiency at the second trap for each set of releases (Figure 22) formed by looking at the total number of recoveries / total fish released.

Figure 22: Empirical total recovery probabilities

There are several unusual features

• There appears to be heterogeneity in the total capture probabilities across the season with a lower catchability early in the season and a higher catchability later in the season
• The fall off in catchability near the end of the season is a artefact of sampling ending while fish are still migrating.

Similarly, let us look at the pattern of unmarked fish captured at the second trap (Figure 23).

Figure 23: Observed number of unmarked recaptures

Again notice no untagged recoveries in weeks 23 to 26 and the large jump in recoveries.

8.3.5.2 Fitting the basic BTSPAS non-diagonal model

The Petersen package function BTSPAS_NonDiag_fit() is a wrapper to the corresponding function in the BTSPAS package that takes the (modified for bad events) data file, a model for the mean capture probabilities at the second temporal stratum (default is a common mean), and identification of break points in the underlying spline, and then call the function in the BTSPAS package, and formats the returned data structure to match the returning structure for other functions in this package.

If finer control over the fitting process is needed, the BTSPAS package should be called directly – please consult the vignettes that come with the BTSPAS package.

In this case, no sampling was done in weeks 23 to 27 and so we set the capture probability of 0 on these days:

BTSPAS.nondiag.fit1 <- LP_BTSPAS_fit_NonDiag( 
         data=data_btspas_nondiag1,
         p_model=~1,
         InitialSeed=34343,
         logitP.fixed=c(23:26), logitP.fixed.values=rep(-10,length(23:26))
)

The distribution of the posterior sample for the total number unmarked and total abundance is shown in ?@fig-btspas-diag1-postUN-nondiag

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319   Utot 320006
320   Utot 330786
321   Utot 392599
322   Utot 347924
323   Utot 331530
324   Utot 330386
325   Utot 375478
326   Utot 316026
327   Utot 363626
328   Utot 309211
329   Utot 300997
330   Utot 380414
331   Utot 311080
332   Utot 314026
333   Utot 307281
334   Utot 354844
335   Utot 349266
336   Utot 347486
337   Utot 338443
338   Utot 289549
339   Utot 294284
340   Utot 315483
341   Utot 357617
342   Utot 307013
343   Utot 340270
344   Utot 352670
345   Utot 377886
346   Utot 301938
347   Utot 351252
348   Utot 316633
349   Utot 352709
350   Utot 297398
351   Utot 354860
352   Utot 319610
353   Utot 318231
354   Utot 379243
355   Utot 296255
356   Utot 308342
357   Utot 306482
358   Utot 346035
359   Utot 313010
360   Utot 348886
361   Utot 338808
362   Utot 312381
363   Utot 312440
364   Utot 381934
365   Utot 318910
366   Utot 296970
367   Utot 336074
368   Utot 275943
369   Utot 320739
370   Utot 398233
371   Utot 313478
372   Utot 355187
373   Utot 380739
374   Utot 368593
375   Utot 332411
376   Utot 323459
377   Utot 375821
378   Utot 373232
379   Utot 347947
380   Utot 328176
381   Utot 301313
382   Utot 347490
383   Utot 273388
384   Utot 307322
385   Utot 348564
386   Utot 335528
387   Utot 320191
388   Utot 310331
389   Utot 334338
390   Utot 301930
391   Utot 293291
392   Utot 346079
393   Utot 320373
394   Utot 342836
395   Utot 301502
396   Utot 287781
397   Utot 297225
398   Utot 356836
399   Utot 319261
400   Utot 302193
401   Utot 363903
402   Utot 300148
403   Utot 289831
404   Utot 326072
405   Utot 417734
406   Utot 402836
407   Utot 326952
408   Utot 308353
409   Utot 315924
410   Utot 309329
411   Utot 322994
412   Utot 296472
413   Utot 309758
414   Utot 316383
415   Utot 297382
416   Utot 365440
417   Utot 336819
418   Utot 289159
419   Utot 335299
420   Utot 419298
421   Utot 321590
422   Utot 348984
423   Utot 279434
424   Utot 277101
425   Utot 308477
426   Utot 355080
427   Utot 300958
428   Utot 330363
429   Utot 312983
430   Utot 434493
431   Utot 286962
432   Utot 358662
433   Utot 327596
434   Utot 346820
435   Utot 313802
436   Utot 301431
437   Utot 360072
438   Utot 365611
439   Utot 338155
440   Utot 368526
441   Utot 340315
442   Utot 334797
443   Utot 333438
444   Utot 321177
445   Utot 298079
446   Utot 312539
447   Utot 309117
448   Utot 384357
449   Utot 281671
450   Utot 376998
451   Utot 332907
452   Utot 310817
453   Utot 326403
454   Utot 328416
455   Utot 335372
456   Utot 304447
457   Utot 296532
458   Utot 324709
459   Utot 294977
460   Utot 388308
461   Utot 332641
462   Utot 272856
463   Utot 283303
464   Utot 332599
465   Utot 367609
466   Utot 321384
467   Utot 336678
468   Utot 386199
469   Utot 340905
470   Utot 301254
471   Utot 330404
472   Utot 322051
473   Utot 345762
474   Utot 389315
475   Utot 308862
476   Utot 309747
477   Utot 290952
478   Utot 395840
479   Utot 349867
480   Utot 323517
481   Utot 273214
482   Utot 350986
483   Utot 300220
484   Utot 327333
485   Utot 281788
486   Utot 316123
487   Utot 410109
488   Utot 330726
489   Utot 328329
490   Utot 412562
491   Utot 359848
492   Utot 359903
493   Utot 343698
494   Utot 319356
495   Utot 375255
496   Utot 288387
497   Utot 341241
498   Utot 351763
499   Utot 355588
500   Utot 310146
501   Utot 353058
502   Utot 311767
503   Utot 406724
504   Utot 296380
505   Utot 308532
506   Utot 317707
507   Utot 322023
508   Utot 330124
509   Utot 357901
510   Utot 351824
511   Utot 317363
512   Utot 331055
513   Utot 305102
514   Utot 346408
515   Utot 320887
516   Utot 343254
517   Utot 277220
518   Utot 284342
519   Utot 334268
520   Utot 319478
521   Utot 369290
522   Utot 336253
523   Utot 355476
524   Utot 366577
525   Utot 340087
526   Utot 301063
527   Utot 292817
528   Utot 301247
529   Utot 309092
530   Utot 344435
531   Utot 288858
532   Utot 412467
533   Utot 386057
534   Utot 331718
535   Utot 311718
536   Utot 317622
537   Utot 286507
538   Utot 355058
539   Utot 332501
540   Utot 322781
541   Utot 346277
542   Utot 331242
543   Utot 283405
544   Utot 312409
545   Utot 345333
546   Utot 329566
547   Utot 278062
548   Utot 308659
549   Utot 327816
550   Utot 478385
551   Utot 356698
552   Utot 304016
553   Utot 314554
554   Utot 327666
555   Utot 339120
556   Utot 361734
557   Utot 296510
558   Utot 308784
559   Utot 332201
560   Utot 334087
561   Utot 372551
562   Utot 352314
563   Utot 348280
564   Utot 296419
565   Utot 323765
566   Utot 300286
567   Utot 376279
568   Utot 359143
569   Utot 301410
570   Utot 268970
571   Utot 307975
572   Utot 324414
573   Utot 357009
574   Utot 358840
575   Utot 346616
576   Utot 291020
577   Utot 333903
578   Utot 298911
579   Utot 350791
580   Utot 321036
581   Utot 280219
582   Utot 347972
583   Utot 336967
584   Utot 294584
585   Utot 340077
586   Utot 346370
587   Utot 374251
588   Utot 289896
589   Utot 316328
590   Utot 298875
591   Utot 290544
592   Utot 312877
593   Utot 292344
594   Utot 308856
595   Utot 317226
596   Utot 348636
597   Utot 317422
598   Utot 381905
599   Utot 330575
600   Utot 359053
601   Utot 317504
602   Utot 309413
603   Utot 353208
604   Utot 368936
605   Utot 294171
606   Utot 344452
607   Utot 301932
608   Utot 324542
609   Utot 307380
610   Utot 332291
611   Utot 369443
612   Utot 351818
613   Utot 350743
614   Utot 302107
615   Utot 303901
616   Utot 367738
617   Utot 321478
618   Utot 287397
619   Utot 330758
620   Utot 315806
621   Utot 353741
622   Utot 374146
623   Utot 290200
624   Utot 357813
625   Utot 302362
626   Utot 342482
627   Utot 360121
628   Utot 331310
629   Utot 349712
630   Utot 293376
631   Utot 351261
632   Utot 318108
633   Utot 351334
634   Utot 348021
635   Utot 306435
636   Utot 352747
637   Utot 329444
638   Utot 332616
639   Utot 316560
640   Utot 302819
641   Utot 316976
642   Utot 325826
643   Utot 305788
644   Utot 306224
645   Utot 305516
646   Utot 303459
647   Utot 320408
648   Utot 396977
649   Utot 350484
650   Utot 306591
651   Utot 296920
652   Utot 313845
653   Utot 392883
654   Utot 318499
655   Utot 374068
656   Utot 304303
657   Utot 335556
658   Utot 358500
659   Utot 439522
660   Utot 340855
661   Utot 407713
662   Utot 298447
663   Utot 284169
664   Utot 316530
665   Utot 296335
666   Utot 317497
667   Utot 326967
668   Utot 346862
669   Utot 302557
670   Utot 346984
671   Utot 350372
672   Utot 346225
673   Utot 288313
674   Utot 349212
675   Utot 268323
676   Utot 397774
677   Utot 316244
678   Utot 356290
679   Utot 339085
680   Utot 305708
681   Utot 348557
682   Utot 313633
683   Utot 316772
684   Utot 346896
685   Utot 382229
686   Utot 300526
687   Utot 329707
688   Utot 295608
689   Utot 282500
690   Utot 323761
691   Utot 337541
692   Utot 319268
693   Utot 344590
694   Utot 285643
695   Utot 298314
696   Utot 288080
697   Utot 332757
698   Utot 288913
699   Utot 304213
700   Utot 324350
701   Utot 395416
702   Utot 313116
703   Utot 310592
704   Utot 361903
705   Utot 295822
706   Utot 301091
707   Utot 386929
708   Utot 302717
709   Utot 385252
710   Utot 368396
711   Utot 319329
712   Utot 288957
713   Utot 316061
714   Utot 282140
715   Utot 312006
716   Utot 312863
717   Utot 368392
718   Utot 376642
719   Utot 314810
720   Utot 267306
721   Utot 358626
722   Utot 391797
723   Utot 328608
724   Utot 304538
725   Utot 318226
726   Utot 307119
727   Utot 324527
728   Utot 381265
729   Utot 302549
730   Utot 329077
731   Utot 276297
732   Utot 344700
733   Utot 436618
734   Utot 413946
735   Utot 314736
736   Utot 374029
737   Utot 313027
738   Utot 320542
739   Utot 327793
740   Utot 329862
741   Utot 346919
742   Utot 391360
743   Utot 320916
744   Utot 316157
745   Utot 341447
746   Utot 297918
747   Utot 299106
748   Utot 362888
749   Utot 348369
750   Utot 334634
751   Utot 275615
752   Utot 303231
753   Utot 342058
754   Utot 297046
755   Utot 326970
756   Utot 316628
757   Utot 356975
758   Utot 389785
759   Utot 307242
760   Utot 374651
761   Utot 352947
762   Utot 335670
763   Utot 330937
764   Utot 384688
765   Utot 337826
766   Utot 343369
767   Utot 381799
768   Utot 308185
769   Utot 389301
770   Utot 332925
771   Utot 400250
772   Utot 331320
773   Utot 284694
774   Utot 307295
775   Utot 346327
776   Utot 331561
777   Utot 404480
778   Utot 305252
779   Utot 337568
780   Utot 340176
781   Utot 390684
782   Utot 302801
783   Utot 408926
784   Utot 333643
785   Utot 314701
786   Utot 305385
787   Utot 379287
788   Utot 295715
789   Utot 358187
790   Utot 355153
791   Utot 327375
792   Utot 338678
793   Utot 340173
794   Utot 319378
795   Utot 282946
796   Utot 304750
797   Utot 281436
798   Utot 293768
799   Utot 339336
800   Utot 322584
801   Utot 327417
802   Utot 331553
803   Utot 375631
804   Utot 305257
805   Utot 339696
806   Utot 314122
807   Utot 299194
808   Utot 303548
809   Utot 311253
810   Utot 319962
811   Utot 272025
812   Utot 292409
813   Utot 286334
814   Utot 311218
815   Utot 309629
816   Utot 358824
817   Utot 337441
818   Utot 332652
819   Utot 347252
820   Utot 389995
821   Utot 320841
822   Utot 403404
823   Utot 342892
824   Utot 318548
825   Utot 296166
826   Utot 312460
827   Utot 329602
828   Utot 347960
829   Utot 380998
830   Utot 350134
831   Utot 276547
832   Utot 304621
833   Utot 361416
834   Utot 318311
835   Utot 333640
836   Utot 354044
837   Utot 313063
838   Utot 309742
839   Utot 317871
840   Utot 335704
841   Utot 407012
842   Utot 307932
843   Utot 353616
844   Utot 365689
845   Utot 296679
846   Utot 406365
847   Utot 341621
848   Utot 291676
849   Utot 355913
850   Utot 314778
851   Utot 316927
852   Utot 307418
853   Utot 336977
854   Utot 290032
855   Utot 361159
856   Utot 315604
857   Utot 342099
858   Utot 332917
859   Utot 325615
860   Utot 298991
861   Utot 355009
862   Utot 375356
863   Utot 280373
864   Utot 413801
865   Utot 283217
866   Utot 309889
867   Utot 438801
868   Utot 297801
869   Utot 318330
870   Utot 312886
871   Utot 423872
872   Utot 329678
873   Utot 380280
874   Utot 307724
875   Utot 342685
876   Utot 320868
877   Utot 360337
878   Utot 297750
879   Utot 353702
880   Utot 340989
881   Utot 402729
882   Utot 361005
883   Utot 371746
884   Utot 321632
885   Utot 309033
886   Utot 304891
887   Utot 348370
888   Utot 343873
889   Utot 291116
890   Utot 331914
891   Utot 324543
892   Utot 287045
893   Utot 319954
894   Utot 311521
895   Utot 345593
896   Utot 346595
897   Utot 441005
898   Utot 339302
899   Utot 310320
900   Utot 319108
901   Utot 316563
902   Utot 322936
903   Utot 337184
904   Utot 389775
905   Utot 289126
906   Utot 371411
907   Utot 339998
908   Utot 318017
909   Utot 342114
910   Utot 310236
911   Utot 345645
912   Utot 345826
913   Utot 353216
914   Utot 321934
915   Utot 308125
916   Utot 362231
917   Utot 421541
918   Utot 364283
919   Utot 423485
920   Utot 338597
921   Utot 312956
922   Utot 303811
923   Utot 358251
924   Utot 317162
925   Utot 327833
926   Utot 330354
927   Utot 351691
928   Utot 302366
929   Utot 335155
930   Utot 349374
931   Utot 344781
932   Utot 366383
933   Utot 329840
934   Utot 398481
935   Utot 323408
936   Utot 375116
937   Utot 339559
938   Utot 322691
939   Utot 363537
940   Utot 335296
941   Utot 325539
942   Utot 292415
943   Utot 317486
944   Utot 330606
945   Utot 314060
946   Utot 344286
947   Utot 313256
948   Utot 333239
949   Utot 304178
950   Utot 358103
951   Utot 343999
952   Utot 282542
953   Utot 297722
954   Utot 339005
955   Utot 356022
956   Utot 313702
957   Utot 303457
958   Utot 318027
959   Utot 373734
960   Utot 373391
961   Utot 328561
962   Utot 317117
963   Utot 291596
964   Utot 310625
965   Utot 312804
966   Utot 315456
967   Utot 350458
968   Utot 326698
969   Utot 308886
970   Utot 287633
971   Utot 327122
972   Utot 316454
973   Utot 292689
974   Utot 308165
975   Utot 357636
976   Utot 310488
977   Utot 324594
978   Utot 285053
979   Utot 330403
980   Utot 325085
981   Utot 322574
982   Utot 330550
983   Utot 450020
984   Utot 365481
985   Utot 355298
986   Utot 295246
987   Utot 308169
988   Utot 331011
989   Utot 355751
990   Utot 381367
991   Utot 312589
992   Utot 374008
993   Utot 348287
994   Utot 334115
995   Utot 317078
996   Utot 324782
997   Utot 309373
998   Utot 288188
999   Utot 355262
1000  Utot 303189
1001  Utot 344214
1002  Utot 288603
1003  Utot 310712
1004  Utot 359820
1005  Utot 346036
1006  Utot 369838
1007  Utot 292026
1008  Utot 395675
1009  Utot 349911
1010  Utot 341689
1011  Utot 359956
1012  Utot 304653
1013  Utot 323592
1014  Utot 328898
1015  Utot 303955
1016  Utot 324288
1017  Utot 298564
1018  Utot 329259
1019  Utot 330137
1020  Utot 319125
1021  Utot 312494
1022  Utot 378053
1023  Utot 319384
1024  Utot 351258
1025  Utot 359608
1026  Utot 296663
1027  Utot 376865
1028  Utot 281571
1029  Utot 352237
1030  Utot 293784
1031  Utot 290060
1032  Utot 366945
1033  Utot 330976
1034  Utot 274448
1035  Utot 364905
1036  Utot 359193
1037  Utot 365159
1038  Utot 345160
1039  Utot 343257
1040  Utot 386009
1041  Utot 399160
1042  Utot 316133
1043  Utot 318830
1044  Utot 287560
1045  Utot 365334
1046  Utot 372966
1047  Utot 283353
1048  Utot 308042
1049  Utot 351382
1050  Utot 383890
1051  Utot 284786
1052  Utot 366086
1053  Utot 317944
1054  Utot 312339
1055  Utot 347034
1056  Utot 333968
1057  Utot 305539
1058  Utot 281262
1059  Utot 295508
1060  Utot 293647
1061  Utot 310809
1062  Utot 329921
1063  Utot 354288
1064  Utot 354294
1065  Utot 357179
1066  Utot 306242
1067  Utot 310318
1068  Utot 330716
1069  Utot 378025
1070  Utot 325001
1071  Utot 328759
1072  Utot 384693
1073  Utot 388725
1074  Utot 311880
1075  Utot 304335
1076  Utot 332389
1077  Utot 326693
1078  Utot 301551
1079  Utot 330699
1080  Utot 359041
1081  Utot 408574
1082  Utot 374628
1083  Utot 313828
1084  Utot 357380
1085  Utot 305732
1086  Utot 300781
1087  Utot 298975
1088  Utot 353301
1089  Utot 335300
1090  Utot 313248
1091  Utot 310172
1092  Utot 326928
1093  Utot 369095
1094  Utot 353247
1095  Utot 410266
1096  Utot 356659
1097  Utot 339229
1098  Utot 374166
1099  Utot 287351
1100  Utot 313760
1101  Utot 286878
1102  Utot 340339
1103  Utot 304952
1104  Utot 280010
1105  Utot 276973
1106  Utot 311202
1107  Utot 323859
1108  Utot 292500
1109  Utot 308262
1110  Utot 336740
1111  Utot 364521
1112  Utot 329898
1113  Utot 336818
1114  Utot 272018
1115  Utot 317798
1116  Utot 341178
1117  Utot 327541
1118  Utot 428821
1119  Utot 357067
1120  Utot 299433
1121  Utot 288476
1122  Utot 334601
1123  Utot 368007
1124  Utot 316873
1125  Utot 286693
1126  Utot 332312
1127  Utot 383679
1128  Utot 358093
1129  Utot 293872
1130  Utot 327213
1131  Utot 341256
1132  Utot 308660
1133  Utot 292111
1134  Utot 375122
1135  Utot 285821
1136  Utot 371876
1137  Utot 327555
1138  Utot 300531
1139  Utot 290157
1140  Utot 310251
1141  Utot 332841
1142  Utot 401595
1143  Utot 327019
1144  Utot 313082
1145  Utot 302456
1146  Utot 402486
1147  Utot 314192
1148  Utot 346051
1149  Utot 300985
1150  Utot 305564
1151  Utot 382319
1152  Utot 302498
1153  Utot 328232
1154  Utot 293504
1155  Utot 297124
1156  Utot 327040
1157  Utot 311573
1158  Utot 328839
1159  Utot 340445
1160  Utot 368068
1161  Utot 390384
1162  Utot 339089
1163  Utot 335376
1164  Utot 307982
1165  Utot 351805
1166  Utot 337692
1167  Utot 312558
1168  Utot 365652
1169  Utot 340805
1170  Utot 306024
1171  Utot 277383
1172  Utot 342600
1173  Utot 294543
1174  Utot 334977
1175  Utot 317448
1176  Utot 300516
1177  Utot 292242
1178  Utot 326625
1179  Utot 346140
1180  Utot 340289
1181  Utot 351679
1182  Utot 366979
1183  Utot 314232
1184  Utot 303626
1185  Utot 402030
1186  Utot 321006
1187  Utot 331874
1188  Utot 261914
1189  Utot 340200
1190  Utot 322221
1191  Utot 336858
1192  Utot 321867
1193  Utot 289619
1194  Utot 296842
1195  Utot 309249
1196  Utot 342002
1197  Utot 422342
1198  Utot 311751
1199  Utot 333512
1200  Utot 340480
1201  Utot 378825
1202  Utot 357409
1203  Utot 387852
1204  Utot 332794
1205  Utot 302508
1206  Utot 337457
1207  Utot 348632
1208  Utot 345693
1209  Utot 344636
1210  Utot 288853
1211  Utot 317508
1212  Utot 321369
1213  Utot 314156
1214  Utot 323102
1215  Utot 329243
1216  Utot 292007
1217  Utot 335587
1218  Utot 295193
1219  Utot 376391
1220  Utot 324945
1221  Utot 272792
1222  Utot 319061
1223  Utot 349202
1224  Utot 353244
1225  Utot 385545
1226  Utot 311909
1227  Utot 330150
1228  Utot 345927
1229  Utot 313221
1230  Utot 283741
1231  Utot 340964
1232  Utot 308455
1233  Utot 296630
1234  Utot 348779
1235  Utot 326952
1236  Utot 296253
1237  Utot 281506
1238  Utot 377298
1239  Utot 323743
1240  Utot 342067
1241  Utot 285007
1242  Utot 306312
1243  Utot 347643
1244  Utot 352307
1245  Utot 420802
1246  Utot 372391
1247  Utot 332411
1248  Utot 348033
1249  Utot 324207
1250  Utot 322504
1251  Utot 316863
1252  Utot 360825
1253  Utot 305868
1254  Utot 347475
1255  Utot 283129
1256  Utot 377930
1257  Utot 328168
1258  Utot 289827
1259  Utot 321609
1260  Utot 358398
1261  Utot 338928
1262  Utot 372177
1263  Utot 331329
1264  Utot 302064
1265  Utot 284270
1266  Utot 294399
1267  Utot 280731
1268  Utot 298408
1269  Utot 366320
1270  Utot 268875
1271  Utot 391320
1272  Utot 303269
1273  Utot 303694
1274  Utot 300323
1275  Utot 353074
1276  Utot 367608
1277  Utot 289388
1278  Utot 372355
1279  Utot 292984
1280  Utot 327560
1281  Utot 319884
1282  Utot 315832
1283  Utot 286306
1284  Utot 304699
1285  Utot 389496
1286  Utot 345807
1287  Utot 312583
1288  Utot 299951
1289  Utot 312605
1290  Utot 330148
1291  Utot 320557
1292  Utot 292258
1293  Utot 353678
1294  Utot 342809
1295  Utot 328482
1296  Utot 375509
1297  Utot 319614
1298  Utot 326464
1299  Utot 368742
1300  Utot 329384
1301  Utot 409390
1302  Utot 317039
1303  Utot 305185
1304  Utot 353565
1305  Utot 372648
1306  Utot 310426
1307  Utot 341656
1308  Utot 295240
1309  Utot 340207
1310  Utot 342220
1311  Utot 292510
1312  Utot 295277
1313  Utot 342768
1314  Utot 325911
1315  Utot 367367
1316  Utot 335312
1317  Utot 329638
1318  Utot 330782
1319  Utot 301586
1320  Utot 389732
1321  Utot 306081
1322  Utot 319051
1323  Utot 311339
1324  Utot 385351
1325  Utot 330868
1326  Utot 327116
1327  Utot 309750
1328  Utot 368080
1329  Utot 313707
1330  Utot 317811
1331  Utot 392539
1332  Utot 350969
1333  Utot 330210
1334  Utot 315486
1335  Utot 342236
1336  Utot 340475
1337  Utot 333332
1338  Utot 313591
1339  Utot 305501
1340  Utot 350915
1341  Utot 392792
1342  Utot 296440
1343  Utot 308345
1344  Utot 274971
1345  Utot 292645
1346  Utot 369564
1347  Utot 339308
1348  Utot 375671
1349  Utot 367585
1350  Utot 343790
1351  Utot 327141
1352  Utot 321587
1353  Utot 368021
1354  Utot 344058
1355  Utot 296360
1356  Utot 396497
1357  Utot 401412
1358  Utot 342123
1359  Utot 292450
1360  Utot 348835
1361  Utot 310608
1362  Utot 316448
1363  Utot 313718
1364  Utot 305776
1365  Utot 325232
1366  Utot 310827
1367  Utot 287567
1368  Utot 293422
1369  Utot 301816
1370  Utot 326750
1371  Utot 308386
1372  Utot 332468
1373  Utot 266474
1374  Utot 381979
1375  Utot 374316
1376  Utot 302811
1377  Utot 362264
1378  Utot 378318
1379  Utot 311800
1380  Utot 374565
1381  Utot 347341
1382  Utot 301608
1383  Utot 320164
1384  Utot 386993
1385  Utot 323326
1386  Utot 362520
1387  Utot 380839
1388  Utot 318965
1389  Utot 381866
1390  Utot 392334
1391  Utot 407093
1392  Utot 324045
1393  Utot 322854
1394  Utot 319427
1395  Utot 382835
1396  Utot 302393
1397  Utot 361537
1398  Utot 316304
1399  Utot 351636
1400  Utot 298040
1401  Utot 311439
1402  Utot 450002
1403  Utot 296231
1404  Utot 355134
1405  Utot 355314
1406  Utot 302423
1407  Utot 297124
1408  Utot 322926
1409  Utot 300501
1410  Utot 317627
1411  Utot 295934
1412  Utot 341557
1413  Utot 331871
1414  Utot 356634
1415  Utot 342822
1416  Utot 338444
1417  Utot 310188
1418  Utot 282479
1419  Utot 346899
1420  Utot 316506
1421  Utot 314806
1422  Utot 386738
1423  Utot 332971
1424  Utot 324480
1425  Utot 377631
1426  Utot 299973
1427  Utot 336214
1428  Utot 296754
1429  Utot 298020
1430  Utot 311261
1431  Utot 292703
1432  Utot 316612
1433  Utot 316238
1434  Utot 407698
1435  Utot 292007
1436  Utot 337748
1437  Utot 338169
1438  Utot 411234
1439  Utot 313362
1440  Utot 277418
1441  Utot 323152
1442  Utot 418436
1443  Utot 338097
1444  Utot 346402
1445  Utot 291973
1446  Utot 327544
1447  Utot 349378
1448  Utot 324534
1449  Utot 455838
1450  Utot 351181
1451  Utot 325766
1452  Utot 372442
1453  Utot 322789
1454  Utot 302421
1455  Utot 350797
1456  Utot 364296
1457  Utot 298006
1458  Utot 353001
1459  Utot 284583
1460  Utot 303903
1461  Utot 332403
1462  Utot 322911
1463  Utot 314473
1464  Utot 323077
1465  Utot 299811
1466  Utot 331575
1467  Utot 350386
1468  Utot 309355
1469  Utot 375935
1470  Utot 419841
1471  Utot 333809
1472  Utot 300244
1473  Utot 360650
1474  Utot 355879
1475  Utot 313857
1476  Utot 361778
1477  Utot 305390
1478  Utot 313072
1479  Utot 319252
1480  Utot 340615
1481  Utot 301682
1482  Utot 291514
1483  Utot 407965
1484  Utot 308117
1485  Utot 327734
1486  Utot 318921
1487  Utot 322614
1488  Utot 325440
1489  Utot 293783
1490  Utot 307528
1491  Utot 410489
1492  Utot 321358
1493  Utot 288514
1494  Utot 303779
1495  Utot 440378
1496  Utot 300994
1497  Utot 418082
1498  Utot 290250
1499  Utot 348772
1500  Utot 353618
1501  Utot 299825
1502  Utot 402598
1503  Utot 342403
1504  Utot 327624
1505  Utot 297252
1506  Utot 333718
1507  Utot 340920
1508  Utot 339032
1509  Utot 335963
1510  Utot 301437
1511  Utot 323808
1512  Utot 377417
1513  Utot 407292
1514  Utot 348283
1515  Utot 322853
1516  Utot 299927
1517  Utot 266895
1518  Utot 352206
1519  Utot 333913
1520  Utot 304258
1521  Utot 302009
1522  Utot 320429
1523  Utot 288768
1524  Utot 357220
1525  Utot 307966
1526  Utot 301885
1527  Utot 373083
1528  Utot 310271
1529  Utot 366419
1530  Utot 332195
1531  Utot 366642
1532  Utot 374837
1533  Utot 292613
1534  Utot 302419
1535  Utot 371274
1536  Utot 317047
1537  Utot 305315
1538  Utot 367416
1539  Utot 295675
1540  Utot 323022
1541  Utot 336898
1542  Utot 283288
1543  Utot 355455
1544  Utot 341477
1545  Utot 329934
1546  Utot 292587
1547  Utot 340386
1548  Utot 290456
1549  Utot 391523
1550  Utot 372479
1551  Utot 364877
1552  Utot 279062
1553  Utot 314971
1554  Utot 314447
1555  Utot 333083
1556  Utot 326265
1557  Utot 292279
1558  Utot 320376
1559  Utot 415394
1560  Utot 314695
1561  Utot 330889
1562  Utot 297403
1563  Utot 330500
1564  Utot 323925
1565  Utot 327422
1566  Utot 345530
1567  Utot 357440
1568  Utot 371769
1569  Utot 430443
1570  Utot 323790
1571  Utot 320225
1572  Utot 305904
1573  Utot 310537
1574  Utot 331431
1575  Utot 323918
1576  Utot 301749
1577  Utot 293506
1578  Utot 430766
1579  Utot 282495
1580  Utot 264637
1581  Utot 366528
1582  Utot 365226
1583  Utot 363083
1584  Utot 332901
1585  Utot 299070
1586  Utot 329639
1587  Utot 299343
1588  Utot 320110
1589  Utot 309847
1590  Utot 398976
1591  Utot 354238
1592  Utot 290964
1593  Utot 331841
1594  Utot 316735
1595  Utot 328232
1596  Utot 329118
1597  Utot 324217
1598  Utot 297090
1599  Utot 327696
1600  Utot 378867
1601  Utot 298357
1602  Utot 334609
1603  Utot 323733
1604  Utot 322359
1605  Utot 349110
1606  Utot 307067
1607  Utot 334342
1608  Utot 330975
1609  Utot 406463
1610  Utot 337936
1611  Utot 345582
1612  Utot 389417
1613  Utot 330691
1614  Utot 331694
1615  Utot 342299
1616  Utot 336172
1617  Utot 346056
1618  Utot 323321
1619  Utot 357539
1620  Utot 298529
1621  Utot 313687
1622  Utot 319782
1623  Utot 340091
1624  Utot 384843
1625  Utot 318903
1626  Utot 411067
1627  Utot 311875
1628  Utot 323677
1629  Utot 332846
1630  Utot 344094
1631  Utot 327224
1632  Utot 324898
1633  Utot 291426
1634  Utot 337719
1635  Utot 347398
1636  Utot 335005
1637  Utot 328396
1638  Utot 278783
1639  Utot 268003
1640  Utot 309351
1641  Utot 354807
1642  Utot 301436
1643  Utot 375096
1644  Utot 378733
1645  Utot 325016
1646  Utot 303327
1647  Utot 342848
1648  Utot 341430
1649  Utot 325368
1650  Utot 328610
1651  Utot 331699
1652  Utot 304197
1653  Utot 320978
1654  Utot 311424
1655  Utot 334783
1656  Utot 324217
1657  Utot 331344
1658  Utot 314509
1659  Utot 288590
1660  Utot 372900
1661  Utot 344777
1662  Utot 332842
1663  Utot 400942
1664  Utot 320171
1665  Utot 299133
1666  Utot 307128
1667  Utot 375790
1668  Utot 355009
1669  Utot 329351
1670  Utot 313354
1671  Utot 332719
1672  Utot 337277
1673  Utot 358679
1674  Utot 354985
1675  Utot 376716
1676  Utot 349743
1677  Utot 294449
1678  Utot 305366
1679  Utot 356874
1680  Utot 348145
1681  Utot 327103
1682  Utot 319597
1683  Utot 419983
1684  Utot 338534
1685  Utot 323237
1686  Utot 313938
1687  Utot 329002
1688  Utot 294572
1689  Utot 325898
1690  Utot 364488
1691  Utot 300276
1692  Utot 317175
1693  Utot 316406
1694  Utot 301934
1695  Utot 367284
1696  Utot 393470
1697  Utot 341451
1698  Utot 318835
1699  Utot 340804
1700  Utot 277092
1701  Utot 299403
1702  Utot 417917
1703  Utot 281547
1704  Utot 326280
1705  Utot 420786
1706  Utot 339845
1707  Utot 304439
1708  Utot 347800
1709  Utot 342240
1710  Utot 325664
1711  Utot 335009
1712  Utot 333167
1713  Utot 294499
1714  Utot 337917
1715  Utot 334350
1716  Utot 323001
1717  Utot 285075
1718  Utot 289087
1719  Utot 306716
1720  Utot 275228
1721  Utot 308569
1722  Utot 303262
1723  Utot 351330
1724  Utot 447922
1725  Utot 306706
1726  Utot 385049
1727  Utot 350497
1728  Utot 313867
1729  Utot 288348
1730  Utot 361246
1731  Utot 319817
1732  Utot 309598
1733  Utot 389079
1734  Utot 304663
1735  Utot 359632
1736  Utot 353315
1737  Utot 344876
1738  Utot 292444
1739  Utot 322099
1740  Utot 285141
1741  Utot 345855
1742  Utot 325372
1743  Utot 402988
1744  Utot 299909
1745  Utot 377279
1746  Utot 314292
1747  Utot 308788
1748  Utot 311603
1749  Utot 302073
1750  Utot 340130
1751  Utot 375988
1752  Utot 330798
1753  Utot 316774
1754  Utot 345692
1755  Utot 330244
1756  Utot 302934
1757  Utot 308743
1758  Utot 357639
1759  Utot 380509
1760  Utot 302357
1761  Utot 304404
1762  Utot 331952
1763  Utot 371708
1764  Utot 303042
1765  Utot 326193
1766  Utot 350743
1767  Utot 347600
1768  Utot 299077
1769  Utot 401869
1770  Utot 349639
1771  Utot 396412
1772  Utot 428775
1773  Utot 332302
1774  Utot 313828
1775  Utot 335910
1776  Utot 407867
1777  Utot 327166
1778  Utot 305360
1779  Utot 281533
1780  Utot 301704
1781  Utot 398094
1782  Utot 328198
1783  Utot 298381
1784  Utot 322103
1785  Utot 295385
1786  Utot 354454
1787  Utot 322662
1788  Utot 335550
1789  Utot 370793
1790  Utot 424372
1791  Utot 301436
1792  Utot 326901
1793  Utot 334768
1794  Utot 378615
1795  Utot 315496
1796  Utot 377388
1797  Utot 328814
1798  Utot 354673
1799  Utot 379266
1800  Utot 268008
1801  Utot 364953
1802  Utot 286644
1803  Utot 375988
1804  Utot 345916
1805  Utot 376961
1806  Utot 308952
1807  Utot 285995
1808  Utot 289460
1809  Utot 283948
1810  Utot 303479
1811  Utot 369373
1812  Utot 354133
1813  Utot 332102
1814  Utot 303272
1815  Utot 265817
1816  Utot 328761
1817  Utot 339406
1818  Utot 341342
1819  Utot 340927
1820  Utot 326886
1821  Utot 333542
1822  Utot 410802
1823  Utot 345599
1824  Utot 285119
1825  Utot 340514
1826  Utot 328752
1827  Utot 375371
1828  Utot 336022
1829  Utot 320706
1830  Utot 386126
1831  Utot 324332
1832  Utot 306838
1833  Utot 308409
1834  Utot 368354
1835  Utot 340923
1836  Utot 368883
1837  Utot 303314
1838  Utot 282133
1839  Utot 301043
1840  Utot 326282
1841  Utot 333954
1842  Utot 322538
1843  Utot 317929
1844  Utot 329718
1845  Utot 304789
1846  Utot 395210
1847  Utot 358597
1848  Utot 338512
1849  Utot 307226
1850  Utot 318770
1851  Utot 304809
1852  Utot 289856
1853  Utot 322346
1854  Utot 386240
1855  Utot 326256
1856  Utot 319726
1857  Utot 297741
1858  Utot 328490
1859  Utot 319223
1860  Utot 336804
1861  Utot 329180
1862  Utot 311957
1863  Utot 320011
1864  Utot 344672
1865  Utot 416517
1866  Utot 381115
1867  Utot 357071
1868  Utot 294784
1869  Utot 325817
1870  Utot 297993
1871  Utot 340681
1872  Utot 338282
1873  Utot 333401
1874  Utot 298318
1875  Utot 310803
1876  Utot 322562
1877  Utot 336796
1878  Utot 286164
1879  Utot 342490
1880  Utot 304594
1881  Utot 289171
1882  Utot 347817
1883  Utot 354177
1884  Utot 313424
1885  Utot 280593
1886  Utot 345050
1887  Utot 313380
1888  Utot 395242
1889  Utot 304410
1890  Utot 360237
1891  Utot 331298
1892  Utot 295810
1893  Utot 294068
1894  Utot 314459
1895  Utot 281285
1896  Utot 296429
1897  Utot 314726
1898  Utot 309265
1899  Utot 398376
1900  Utot 288261
1901  Utot 310609
1902  Utot 346128
1903  Utot 317845
1904  Utot 326823
1905  Utot 325610
1906  Utot 281760
1907  Utot 376010
1908  Utot 421897
1909  Utot 304094
1910  Utot 344502
1911  Utot 323726
1912  Utot 298569
1913  Utot 294596
1914  Utot 337761
1915  Utot 312760
1916  Utot 350442
1917  Utot 311548
1918  Utot 357827
1919  Utot 377342
1920  Utot 380234
1921  Utot 331125
1922  Utot 323710
1923  Utot 300888
1924  Utot 361589
1925  Utot 324555
1926  Utot 338065
1927  Utot 405246
1928  Utot 293167
1929  Utot 306376
1930  Utot 316331
1931  Utot 345321
1932  Utot 444443
1933  Utot 349290
1934  Utot 352774
1935  Utot 314372
1936  Utot 284970
1937  Utot 349490
1938  Utot 368897
1939  Utot 347219
1940  Utot 314740
1941  Utot 349303
1942  Utot 318496
1943  Utot 326897
1944  Utot 337677
1945  Utot 287286
1946  Utot 328394
1947  Utot 367170
1948  Utot 362925
1949  Utot 319190
1950  Utot 317675
1951  Utot 311425
1952  Utot 336930
1953  Utot 355754
1954  Utot 385481
1955  Utot 304853
1956  Utot 285425
1957  Utot 336938
1958  Utot 347372
1959  Utot 331895
1960  Utot 368240
1961  Utot 308980
1962  Utot 294982
1963  Utot 361256
1964  Utot 305256
1965  Utot 327059
1966  Utot 357737
1967  Utot 317075
1968  Utot 361952
1969  Utot 309105
1970  Utot 320291
1971  Utot 330163
1972  Utot 310917
1973  Utot 288812
1974  Utot 332003
1975  Utot 389847
1976  Utot 307349
1977  Utot 360492
1978  Utot 334866
1979  Utot 385825
1980  Utot 287870
1981  Utot 314420
1982  Utot 310086
1983  Utot 293530
1984  Utot 451158
1985  Utot 366631
1986  Utot 360841
1987  Utot 316028
1988  Utot 340885
1989  Utot 383552
1990  Utot 362258
1991  Utot 330413
1992  Utot 349890
1993  Utot 355428
1994  Utot 354100
1995  Utot 307728
1996  Utot 296775
1997  Utot 295638
1998  Utot 341428
1999  Utot 324284
2000  Utot 351005
2001  Utot 294420
2002  Utot 325751
2003  Utot 308685
2004  Utot 290557
2005  Utot 322287
2006  Utot 412734
2007  Utot 318294
2008  Utot 295829
2009  Utot 349822
2010  Utot 362434
2011  Utot 336616
2012  Utot 287269
2013  Utot 262982
2014  Utot 410416
2015  Utot 318096
2016  Utot 321735
2017  Utot 352529
2018  Utot 315389
2019  Utot 324806
2020  Utot 346394
2021  Utot 336366
2022  Utot 316312
2023  Utot 435356
2024  Utot 363733
2025  Utot 297616
2026  Utot 375089
2027  Utot 333691
2028  Utot 311757
2029  Utot 340812
2030  Utot 356787
2031  Utot 365011
2032  Utot 348390
2033  Utot 322265
2034  Utot 321330
2035  Utot 297165
2036  Utot 304464
2037  Utot 346016
2038  Utot 317051
2039  Utot 330126
2040  Utot 331823
2041  Utot 427434
2042  Utot 324863
2043  Utot 302062
2044  Utot 383465
2045  Utot 335796
2046  Utot 349114
2047  Utot 305163
2048  Utot 285481
2049  Utot 319155
2050  Utot 334870
2051  Utot 282957
2052  Utot 348048
2053  Utot 342654
2054  Utot 397012
2055  Utot 338783
2056  Utot 300701
2057  Utot 366377
2058  Utot 301019
2059  Utot 289427
2060  Utot 353527
2061  Utot 332737
2062  Utot 290286
2063  Utot 284602
2064  Utot 334492
2065  Utot 305532
2066  Utot 324454
2067  Utot 371376
2068  Utot 295702
2069  Utot 330844
2070  Utot 329676
2071  Utot 347487
2072  Utot 283232
2073  Utot 358594
2074  Utot 311681
2075  Utot 325267
2076  Utot 362472
2077  Utot 336867
2078  Utot 412271
2079  Utot 297927
2080  Utot 324761
2081  Utot 352605
2082  Utot 363537
2083  Utot 317577
2084  Utot 324173
2085  Utot 309408
2086  Utot 270461
2087  Utot 343790
2088  Utot 318199
2089  Utot 388219
2090  Utot 310889
2091  Utot 301392
2092  Utot 297220
2093  Utot 327117
2094  Utot 309968
2095  Utot 306747
2096  Utot 289779
2097  Utot 331903
2098  Utot 301612
2099  Utot 323991
2100  Utot 322527
2101  Utot 281677
2102  Utot 315020
2103  Utot 349203
2104  Utot 293258
2105  Utot 316291
2106  Utot 291648
2107  Utot 286557
2108  Utot 303153
2109  Utot 330279
2110  Utot 350883
2111  Utot 308447
2112  Utot 322159
2113  Utot 356556
2114  Utot 356280
2115  Utot 267718
2116  Utot 365398
2117  Utot 324683
2118  Utot 299301
2119  Utot 364866
2120  Utot 284872
2121  Utot 391736
2122  Utot 270565
2123  Utot 325811
2124  Utot 401569
2125  Utot 398442
2126  Utot 362101
2127  Utot 347118
2128  Utot 339151
2129  Utot 349494
2130  Utot 347499
2131  Utot 330426
2132  Utot 336434
2133  Utot 359231
2134  Utot 345149
2135  Utot 355479
2136  Utot 364134
2137  Utot 294542
2138  Utot 351760
2139  Utot 306244
2140  Utot 292018
2141  Utot 339757
2142  Utot 355073
2143  Utot 313267
2144  Utot 333690
2145  Utot 302867
2146  Utot 376844
2147  Utot 294281
2148  Utot 319964
2149  Utot 359526
2150  Utot 303707
2151  Utot 364674
2152  Utot 297594
2153  Utot 309286
2154  Utot 443768
2155  Utot 411724
2156  Utot 377676
2157  Utot 322818
2158  Utot 386149
2159  Utot 301994
2160  Utot 344932
2161  Utot 296548
2162  Utot 358665
2163  Utot 318283
2164  Utot 351384
2165  Utot 333774
2166  Utot 330779
2167  Utot 356637
2168  Utot 415738
2169  Utot 311901
2170  Utot 320664
2171  Utot 351961
2172  Utot 345239
2173  Utot 424395
2174  Utot 351898
2175  Utot 346992
2176  Utot 334582
2177  Utot 307559
2178  Utot 292325
2179  Utot 287728
2180  Utot 293255
2181  Utot 286243
2182  Utot 282124
2183  Utot 297374
2184  Utot 371307
2185  Utot 382775
2186  Utot 312564
2187  Utot 308671
2188  Utot 313091
2189  Utot 294857
2190  Utot 312519
2191  Utot 342100
2192  Utot 359000
2193  Utot 379073
2194  Utot 332450
2195  Utot 320717
2196  Utot 307017
2197  Utot 302446
2198  Utot 324614
2199  Utot 299980
2200  Utot 345438
2201  Utot 293009
2202  Utot 339444
2203  Utot 314127
2204  Utot 334541
2205  Utot 290329
2206  Utot 357876
2207  Utot 308127
2208  Utot 308618
2209  Utot 308437
2210  Utot 308821
2211  Utot 291415
2212  Utot 300669
2213  Utot 298220
2214  Utot 323808
2215  Utot 305780
2216  Utot 356949
2217  Utot 334875
2218  Utot 357952
2219  Utot 389575
2220  Utot 356991
2221  Utot 273025
2222  Utot 326558
2223  Utot 308484
2224  Utot 290030
2225  Utot 316019
2226  Utot 325821
2227  Utot 330867
2228  Utot 297444
2229  Utot 302025
2230  Utot 302070
2231  Utot 307003
2232  Utot 300301
2233  Utot 322126
2234  Utot 317568
2235  Utot 356008
2236  Utot 306036
2237  Utot 278731
2238  Utot 286212
2239  Utot 312864
2240  Utot 341969
2241  Utot 351786
2242  Utot 357307
2243  Utot 325934
2244  Utot 325360
2245  Utot 316615
2246  Utot 391850
2247  Utot 330101
2248  Utot 335710
2249  Utot 327606
2250  Utot 301609
2251  Utot 316549
2252  Utot 356192
2253  Utot 309868
2254  Utot 351468
2255  Utot 356278
2256  Utot 296714
2257  Utot 344894
2258  Utot 331528
2259  Utot 348262
2260  Utot 302099
2261  Utot 292562
2262  Utot 310666
2263  Utot 346489
2264  Utot 324871
2265  Utot 410117
2266  Utot 297194
2267  Utot 293533
2268  Utot 385341
2269  Utot 321711
2270  Utot 314296
2271  Utot 341009
2272  Utot 317528
2273  Utot 350810
2274  Utot 288151
2275  Utot 336337
2276  Utot 379722
2277  Utot 365840
2278  Utot 319704
2279  Utot 312930
2280  Utot 286197
2281  Utot 363368
2282  Utot 315759
2283  Utot 327088
2284  Utot 334799
2285  Utot 351255
2286  Utot 362647
2287  Utot 294392
2288  Utot 317360
2289  Utot 304386
2290  Utot 281141
2291  Utot 346274
2292  Utot 283323
2293  Utot 301548
2294  Utot 298057
2295  Utot 326887
2296  Utot 415603
2297  Utot 379446
2298  Utot 357888
2299  Utot 407257
2300  Utot 302909
2301  Utot 357315
2302  Utot 357535
2303  Utot 373895
2304  Utot 338641
2305  Utot 334763
2306  Utot 316342
2307  Utot 375828
2308  Utot 303494
2309  Utot 301754
2310  Utot 327314
2311  Utot 318441
2312  Utot 323989
2313  Utot 328035
2314  Utot 416664
2315  Utot 289839
2316  Utot 352407
2317  Utot 378312
2318  Utot 362126
2319  Utot 328916
2320  Utot 333667
2321  Utot 294200
2322  Utot 340876
2323  Utot 288566
2324  Utot 357067
2325  Utot 315994
2326  Utot 387645
2327  Utot 285343
2328  Utot 340146
2329  Utot 323938
2330  Utot 373832
2331  Utot 337899
2332  Utot 356148
2333  Utot 324131
2334  Utot 406344
2335  Utot 321856
2336  Utot 332332
2337  Utot 315291
2338  Utot 299107
2339  Utot 363350
2340  Utot 291736
2341  Utot 316364
2342  Utot 353517
2343  Utot 293461
2344  Utot 350956
2345  Utot 295160
2346  Utot 308204
2347  Utot 407597
2348  Utot 344111
2349  Utot 345901
2350  Utot 295549
2351  Utot 345685
2352  Utot 344540
2353  Utot 338519
2354  Utot 302417
2355  Utot 296779
2356  Utot 325321
2357  Utot 305732
2358  Utot 338419
2359  Utot 332408
2360  Utot 351895
2361  Utot 316381
2362  Utot 315534
2363  Utot 297712
2364  Utot 340178
2365  Utot 306256
2366  Utot 390943
2367  Utot 356565
2368  Utot 349197
2369  Utot 372024
2370  Utot 310917
2371  Utot 375285
2372  Utot 307348
2373  Utot 419285
2374  Utot 476484
2375  Utot 353559
2376  Utot 311484
2377  Utot 314129
2378  Utot 301752
2379  Utot 319487
2380  Utot 300092
2381  Utot 296767
2382  Utot 283133
2383  Utot 324075
2384  Utot 320986
2385  Utot 307020
2386  Utot 334464
2387  Utot 399582
2388  Utot 307513
2389  Utot 285235
2390  Utot 319938
2391  Utot 344531
2392  Utot 300440
2393  Utot 309951
2394  Utot 277340
2395  Utot 412289
2396  Utot 349777
2397  Utot 310070
2398  Utot 301207
2399  Utot 312370
2400  Utot 340357
2401  Utot 316631
2402  Utot 279655
2403  Utot 343530
2404  Utot 369301
2405  Utot 304123
2406  Utot 308121
2407  Utot 365170
2408  Utot 327560
2409  Utot 285745
2410  Utot 299132
2411  Utot 286687
2412  Utot 316385
2413  Utot 292914
2414  Utot 350548
2415  Utot 325824
2416  Utot 280876
2417  Utot 329218
2418  Utot 299642
2419  Utot 348507
2420  Utot 306622
2421  Utot 340908
2422  Utot 311627
2423  Utot 363953
2424  Utot 320334
2425  Utot 346383
2426  Utot 295049
2427  Utot 324247
2428  Utot 284064
2429  Utot 345860
2430  Utot 309535
2431  Utot 380036
2432  Utot 321673
2433  Utot 344295
2434  Utot 332290
2435  Utot 319206
2436  Utot 317597
2437  Utot 335359
2438  Utot 376673
2439  Utot 373913
2440  Utot 304395
2441  Utot 312520
2442  Utot 298670
2443  Utot 317810
2444  Utot 305980
2445  Utot 349573
2446  Utot 288972
2447  Utot 305697
2448  Utot 292343
2449  Utot 380358
2450  Utot 360865
2451  Utot 322000
2452  Utot 334979
2453  Utot 345486
2454  Utot 301574
2455  Utot 382049
2456  Utot 330974
2457  Utot 367384
2458  Utot 353788
2459  Utot 337319
2460  Utot 326541
2461  Utot 305704
2462  Utot 319697
2463  Utot 297213
2464  Utot 322954
2465  Utot 296552
2466  Utot 325281
2467  Utot 307731
2468  Utot 360441
2469  Utot 303613
2470  Utot 330149
2471  Utot 328412
2472  Utot 302975
2473  Utot 355204
2474  Utot 311249
2475  Utot 367005
2476  Utot 348155
2477  Utot 374402
2478  Utot 367162
2479  Utot 320055
2480  Utot 403928
2481  Utot 331961
2482  Utot 356390
2483  Utot 321698
2484  Utot 342538
2485  Utot 333227
2486  Utot 367606
2487  Utot 310228
2488  Utot 349392
2489  Utot 372238
2490  Utot 381296
2491  Utot 306524
2492  Utot 349279
2493  Utot 343262
2494  Utot 311257
2495  Utot 299487
2496  Utot 314603
2497  Utot 432623
2498  Utot 337705
2499  Utot 340622
2500  Utot 333874
2501  Utot 327032
2502  Utot 354372
2503  Utot 332923
2504  Utot 285615
2505  Utot 331230
2506  Utot 305788
2507  Utot 359772
2508  Utot 315392
2509  Utot 361988
2510  Utot 309968
2511  Utot 331694
2512  Utot 357136
2513  Utot 308074
2514  Utot 326217
2515  Utot 333206
2516  Utot 321333
2517  Utot 345956
2518  Utot 367400
2519  Utot 279977
2520  Utot 309408
2521  Utot 306355
2522  Utot 396534
2523  Utot 336680
2524  Utot 332058
2525  Utot 320811
2526  Utot 335286
2527  Utot 355693
2528  Utot 328720
2529  Utot 299908
2530  Utot 323302
2531  Utot 345531
2532  Utot 345926
2533  Utot 318605
2534  Utot 352268
2535  Utot 372224
2536  Utot 296970
2537  Utot 361236
2538  Utot 314224
2539  Utot 282920
2540  Utot 341616
2541  Utot 359282
2542  Utot 315992
2543  Utot 344122
2544  Utot 302330
2545  Utot 278331
2546  Utot 398390
2547  Utot 341507
2548  Utot 312910
2549  Utot 389677
2550  Utot 379973
2551  Utot 337651
2552  Utot 311340
2553  Utot 355012
2554  Utot 357384
2555  Utot 298752
2556  Utot 309742
2557  Utot 346350
2558  Utot 357619
2559  Utot 297815
2560  Utot 279312
2561  Utot 381545
2562  Utot 376666
2563  Utot 342853
2564  Utot 282129
2565  Utot 331463
2566  Utot 346005
2567  Utot 273292
2568  Utot 324249
2569  Utot 269480
2570  Utot 448914
2571  Utot 349079
2572  Utot 328385
2573  Utot 348065
2574  Utot 380656
2575  Utot 363969
2576  Utot 330960
2577  Utot 352372
2578  Utot 321667
2579  Utot 336897
2580  Utot 398624
2581  Utot 303547
2582  Utot 310982
2583  Utot 386417
2584  Utot 313385
2585  Utot 326863
2586  Utot 352560
2587  Utot 311883
2588  Utot 363299
2589  Utot 369588
2590  Utot 343330
2591  Utot 348475
2592  Utot 317071
2593  Utot 334045
2594  Utot 351029
2595  Utot 306036
2596  Utot 305454
2597  Utot 328396
2598  Utot 316699
2599  Utot 288310
2600  Utot 290060
2601  Utot 338248
2602  Utot 304123
2603  Utot 356624
2604  Utot 322798
2605  Utot 310217
2606  Utot 428281
2607  Utot 329939
2608  Utot 329464
2609  Utot 297981
2610  Utot 299900
2611  Utot 333086
2612  Utot 339697
2613  Utot 338916
2614  Utot 296243
2615  Utot 292888
2616  Utot 319689
2617  Utot 290094
2618  Utot 315339
2619  Utot 365318
2620  Utot 340058
2621  Utot 337985
2622  Utot 364288
2623  Utot 310991
2624  Utot 327937
2625  Utot 322752
2626  Utot 345994
2627  Utot 337038
2628  Utot 329748
2629  Utot 300185
2630  Utot 321292
2631  Utot 371548
2632  Utot 291385
2633  Utot 359773
2634  Utot 332523
2635  Utot 307698
2636  Utot 310165
2637  Utot 350665
2638  Utot 457974
2639  Utot 347319
2640  Utot 327932
2641  Utot 322040
2642  Utot 328220
2643  Utot 336949
2644  Utot 369890
2645  Utot 324745
2646  Utot 347867
2647  Utot 354970
2648  Utot 276917
2649  Utot 295583
2650  Utot 404440
2651  Utot 337846
2652  Utot 364096
2653  Utot 356005
2654  Utot 370422
2655  Utot 334981
2656  Utot 291833
2657  Utot 359173
2658  Utot 287544
2659  Utot 303292
2660  Utot 418610
2661  Utot 305488
2662  Utot 335691
2663  Utot 333985
2664  Utot 306209
2665  Utot 307896
2666  Utot 279545
2667  Utot 351534
2668  Utot 299949
2669  Utot 385540
2670  Utot 310544
2671  Utot 406688
2672  Utot 332085
2673  Utot 344065
2674  Utot 313357
2675  Utot 286596
2676  Utot 434481
2677  Utot 367144
2678  Utot 316988
2679  Utot 321821
2680  Utot 448670
2681  Utot 312296
2682  Utot 383590
2683  Utot 288959
2684  Utot 327462
2685  Utot 349514
2686  Utot 328395
2687  Utot 307326
2688  Utot 396224
2689  Utot 351189
2690  Utot 365509
2691  Utot 323157
2692  Utot 370921
2693  Utot 347205
2694  Utot 334517
2695  Utot 384866
2696  Utot 390435
2697  Utot 395724
2698  Utot 308677
2699  Utot 359752
2700  Utot 302764
2701  Utot 305809
2702  Utot 331011
2703  Utot 321387
2704  Utot 301656
2705  Utot 295932
2706  Utot 331475
2707  Utot 314487
2708  Utot 296313
2709  Utot 335578
2710  Utot 289850
2711  Utot 368361
2712  Utot 298102
2713  Utot 306965
2714  Utot 365488
2715  Utot 359019
2716  Utot 317118
2717  Utot 311528
2718  Utot 351755
2719  Utot 334244
2720  Utot 301279
2721  Utot 321373
2722  Utot 331872
2723  Utot 312352
2724  Utot 307577
2725  Utot 397483
2726  Utot 324201
2727  Utot 316372
2728  Utot 389269
2729  Utot 392427
2730  Utot 318036
2731  Utot 335621
2732  Utot 362748
2733  Utot 322971
2734  Utot 377025
2735  Utot 314083
2736  Utot 427448
2737  Utot 401825
2738  Utot 302785
2739  Utot 384300
2740  Utot 304686
2741  Utot 389220
2742  Utot 383912
2743  Utot 330826
2744  Utot 308045
2745  Utot 362316
2746  Utot 296879
2747  Utot 292195
2748  Utot 423086
2749  Utot 318007
2750  Utot 378460
2751  Utot 336076
2752  Utot 348937
2753  Utot 324299
2754  Utot 312318
2755  Utot 301486
2756  Utot 331183
2757  Utot 324503
2758  Utot 329487
2759  Utot 329308
2760  Utot 324337
2761  Utot 300380
2762  Utot 284035
2763  Utot 307435
2764  Utot 300701
2765  Utot 320293
2766  Utot 324295
2767  Utot 300352
2768  Utot 312637
2769  Utot 386683
2770  Utot 353106
2771  Utot 315387
2772  Utot 401215
2773  Utot 274006
2774  Utot 413334
2775  Utot 447433
2776  Utot 329634
2777  Utot 398977
2778  Utot 305928
2779  Utot 306383
2780  Utot 342082
2781  Utot 293677
2782  Utot 358952
2783  Utot 317530
2784  Utot 342158
2785  Utot 267857
2786  Utot 455906
2787  Utot 306204
2788  Utot 369667
2789  Utot 403073
2790  Utot 380809
2791  Utot 335864
2792  Utot 306906
2793  Utot 290812
2794  Utot 278492
2795  Utot 428684
2796  Utot 324499
2797  Utot 394990
2798  Utot 363705
2799  Utot 327803
2800  Utot 323476
2801  Utot 407097
2802  Utot 307036
2803  Utot 318617
2804  Utot 316777
2805  Utot 323049
2806  Utot 327750
2807  Utot 304361
2808  Utot 340899
2809  Utot 355781
2810  Utot 322863
2811  Utot 396784
2812  Utot 328448
2813  Utot 290679
2814  Utot 402097
2815  Utot 347898
2816  Utot 307049
2817  Utot 297754
2818  Utot 419577
2819  Utot 392806
2820  Utot 333757
2821  Utot 311490
2822  Utot 330883
2823  Utot 304042
2824  Utot 312401
2825  Utot 325522
2826  Utot 401083
2827  Utot 326901
2828  Utot 294541
2829  Utot 299721
2830  Utot 341064
2831  Utot 356882
2832  Utot 334645
2833  Utot 321742
2834  Utot 324504
2835  Utot 338424
2836  Utot 335130
2837  Utot 298131
2838  Utot 306643
2839  Utot 369257
2840  Utot 313351
2841  Utot 317987
2842  Utot 365492
2843  Utot 357205
2844  Utot 307828
2845  Utot 344029
2846  Utot 323810
2847  Utot 379680
2848  Utot 313963
2849  Utot 310491
2850  Utot 356900
2851  Utot 346051
2852  Utot 362870
2853  Utot 303025
2854  Utot 398417
2855  Utot 338707
2856  Utot 386326
2857  Utot 326103
2858  Utot 298597
2859  Utot 292277
2860  Utot 389348
2861  Utot 340688
2862  Utot 304109
2863  Utot 389420
2864  Utot 303204
2865  Utot 376158
2866  Utot 308016
2867  Utot 300853
2868  Utot 381365
2869  Utot 331021
2870  Utot 307916
2871  Utot 328270
2872  Utot 304152
2873  Utot 349338
2874  Utot 343978
2875  Utot 301582
2876  Utot 386999
2877  Utot 316343
2878  Utot 287613
2879  Utot 324413
2880  Utot 342084
2881  Utot 381902
2882  Utot 354023
2883  Utot 332280
2884  Utot 335288
2885  Utot 360645
2886  Utot 374387
2887  Utot 317267
2888  Utot 312716
2889  Utot 298443
2890  Utot 306182
2891  Utot 318463
2892  Utot 379340
2893  Utot 291429
2894  Utot 340257
2895  Utot 344356
2896  Utot 317358
2897  Utot 351550
2898  Utot 363069
2899  Utot 328874
2900  Utot 290728
2901  Utot 303373
2902  Utot 324660
2903  Utot 352722
2904  Utot 347438
2905  Utot 289186
2906  Utot 421970
2907  Utot 311688
2908  Utot 327785
2909  Utot 334496
2910  Utot 320181
2911  Utot 381906
2912  Utot 331946
2913  Utot 312195
2914  Utot 340532
2915  Utot 319226
2916  Utot 311407
2917  Utot 265295
2918  Utot 335904
2919  Utot 382823
2920  Utot 308403
2921  Utot 355459
2922  Utot 339732
2923  Utot 359448
2924  Utot 338219
2925  Utot 298297
2926  Utot 331514
2927  Utot 359874
2928  Utot 374970
2929  Utot 306474
2930  Utot 356319
2931  Utot 299423
2932  Utot 319957
2933  Utot 391460
2934  Utot 332989
2935  Utot 306198
2936  Utot 293734
2937  Utot 345319
2938  Utot 307409
2939  Utot 303682
2940  Utot 353077
2941  Utot 351876
2942  Utot 332884
2943  Utot 341481
2944  Utot 312220
2945  Utot 369389
2946  Utot 316744
2947  Utot 327651
2948  Utot 341810
2949  Utot 319687
2950  Utot 364331
2951  Utot 310131
2952  Utot 287008
2953  Utot 295205
2954  Utot 342210
2955  Utot 351253
2956  Utot 355139
2957  Utot 323624
2958  Utot 303423
2959  Utot 297732
2960  Utot 315539
2961  Utot 392622
2962  Utot 338320
2963  Utot 348380
2964  Utot 320716
2965  Utot 350101
2966  Utot 296926
2967  Utot 370637
2968  Utot 318967
2969  Utot 321977
2970  Utot 353660
2971  Utot 282066
2972  Utot 355708
2973  Utot 301878
2974  Utot 308910
2975  Utot 364580
2976  Utot 290600
2977  Utot 290474
2978  Utot 312680
2979  Utot 353469
2980  Utot 353544
2981  Utot 346706
2982  Utot 303516
2983  Utot 304704
2984  Utot 376727
2985  Utot 334358
2986  Utot 323934
2987  Utot 301255
2988  Utot 371712
2989  Utot 303543
2990  Utot 281768
2991  Utot 365937
2992  Utot 313431
2993  Utot 295733
2994  Utot 341112
2995  Utot 298140
2996  Utot 305757
2997  Utot 325730
2998  Utot 307533
2999  Utot 302739
3000  Utot 363655
3001  Utot 356432
3002  Utot 279722
3003  Utot 378123
3004  Utot 345277
3005  Utot 282637
3006  Utot 326250
3007  Utot 351954
3008  Utot 297872
3009  Utot 358649
3010  Utot 380600
3011  Utot 309284
3012  Utot 347822
3013  Utot 328559
3014  Utot 296614
3015  Utot 298282
3016  Utot 349588
3017  Utot 373187
3018  Utot 291692
3019  Utot 350117
3020  Utot 276915
3021  Utot 338728
3022  Utot 375863
3023  Utot 316293
3024  Utot 308053
3025  Utot 325209
3026  Utot 376278
3027  Utot 276703
3028  Utot 341085
3029  Utot 348277
3030  Utot 352384
3031  Utot 369182
3032  Utot 327018
3033  Utot 327237
3034  Utot 384925
3035  Utot 381472
3036  Utot 318235
3037  Utot 337309
3038  Utot 333631
3039  Utot 340928
3040  Utot 301630
3041  Utot 311142
3042  Utot 376344
3043  Utot 332280
3044  Utot 400402
3045  Utot 331271
3046  Utot 351284
3047  Utot 358612
3048  Utot 324147
3049  Utot 309347
3050  Utot 542497
3051  Utot 325966
3052  Utot 303762
3053  Utot 339459
3054  Utot 274581
3055  Utot 404608
3056  Utot 352534
3057  Utot 292173
3058  Utot 302587
3059  Utot 330452
3060  Utot 288009
3061  Utot 352581
3062  Utot 348950
3063  Utot 286128
3064  Utot 339634
3065  Utot 316846
3066  Utot 326374
3067  Utot 339921
3068  Utot 335427
3069  Utot 366086
3070  Utot 302158
3071  Utot 307845
3072  Utot 322559
3073  Utot 318920
3074  Utot 324497
3075  Utot 313554
3076  Utot 422264
3077  Utot 361324
3078  Utot 320271
3079  Utot 304514
3080  Utot 330746
3081  Utot 361779
3082  Utot 312802
3083  Utot 313269
3084  Utot 329671
3085  Utot 343294
3086  Utot 292483
3087  Utot 309879
3088  Utot 341085
3089  Utot 309959
3090  Utot 343669
3091  Utot 323173
3092  Utot 334006
3093  Utot 338286
3094  Utot 341085
3095  Utot 383030
3096  Utot 339404
3097  Utot 362743
3098  Utot 348192
3099  Utot 362200
3100  Utot 395817
3101  Utot 326958
3102  Utot 333821
3103  Utot 337425
3104  Utot 336894
3105  Utot 332025
3106  Utot 312956
3107  Utot 376160
3108  Utot 348092
3109  Utot 298064
3110  Utot 337183
3111  Utot 326821
3112  Utot 322437
3113  Utot 324089
3114  Utot 312399
3115  Utot 346332
3116  Utot 301295
3117  Utot 407612
3118  Utot 315526
3119  Utot 350808
3120  Utot 305171
3121  Utot 270708
3122  Utot 321061
3123  Utot 343626
3124  Utot 349408
3125  Utot 358147
3126  Utot 379307
3127  Utot 307876
3128  Utot 315994
3129  Utot 341473
3130  Utot 332195
3131  Utot 329543
3132  Utot 297184
3133  Utot 350565
3134  Utot 316479
3135  Utot 290499
3136  Utot 346128
3137  Utot 362436
3138  Utot 306733
3139  Utot 354593
3140  Utot 333476
3141  Utot 403568
3142  Utot 286604
3143  Utot 303657
3144  Utot 352529
3145  Utot 384743
3146  Utot 337013
3147  Utot 339598
3148  Utot 347228
3149  Utot 406492
3150  Utot 356186
3151  Utot 283026
3152  Utot 304464
3153  Utot 312458
3154  Utot 373510
3155  Utot 325645
3156  Utot 288227
3157  Utot 313369
3158  Utot 368165
3159  Utot 279763
3160  Utot 335461
3161  Utot 432928
3162  Utot 332816
3163  Utot 365120
3164  Utot 382422
3165  Utot 324081
3166  Utot 305014
3167  Utot 302938
3168  Utot 378115
3169  Utot 329343
3170  Utot 321697
3171  Utot 366475
3172  Utot 273692
3173  Utot 348011
3174  Utot 279425
3175  Utot 315705
3176  Utot 369029
3177  Utot 336548
3178  Utot 319595
3179  Utot 309194
3180  Utot 326912
3181  Utot 374071
3182  Utot 309876
3183  Utot 309754
3184  Utot 358281
3185  Utot 310408
3186  Utot 350441
3187  Utot 294413
3188  Utot 291355
3189  Utot 379759
3190  Utot 338920
3191  Utot 311359
3192  Utot 299537
3193  Utot 324955
3194  Utot 281939
3195  Utot 293053
3196  Utot 325793
3197  Utot 296337
3198  Utot 269922
3199  Utot 324708
3200  Utot 321148
3201  Utot 359586
3202  Utot 335728
3203  Utot 313335
3204  Utot 339869
3205  Utot 320590
3206  Utot 288340
3207  Utot 354909
3208  Utot 366197
3209  Utot 299305
3210  Utot 334382
3211  Utot 343591
3212  Utot 403668
3213  Utot 285140
3214  Utot 410742
3215  Utot 327168
3216  Utot 350133
3217  Utot 326029
3218  Utot 321610
3219  Utot 327377
3220  Utot 341994
3221  Utot 316807
3222  Utot 343806
3223  Utot 339381
3224  Utot 347769
3225  Utot 451867
3226  Utot 315625
3227  Utot 308671
3228  Utot 310230
3229  Utot 348584
3230  Utot 325596
3231  Utot 282392
3232  Utot 334564
3233  Utot 331866
3234  Utot 311160
3235  Utot 339797
3236  Utot 306148
3237  Utot 303665
3238  Utot 309704
3239  Utot 296318
3240  Utot 320766
3241  Utot 337564
3242  Utot 507633
3243  Utot 320131
3244  Utot 293595
3245  Utot 307914
3246  Utot 360582
3247  Utot 483391
3248  Utot 286223
3249  Utot 345869
3250  Utot 334321
3251  Utot 289595
3252  Utot 321240
3253  Utot 330318
3254  Utot 411721
3255  Utot 368012
3256  Utot 353730
3257  Utot 319001
3258  Utot 287864
3259  Utot 303617
3260  Utot 309689
3261  Utot 345073
3262  Utot 299595
3263  Utot 347194
3264  Utot 290507
3265  Utot 320553
3266  Utot 323956
3267  Utot 365779
3268  Utot 318236
3269  Utot 338801
3270  Utot 351352
3271  Utot 297506
3272  Utot 340824
3273  Utot 301519
3274  Utot 296073
3275  Utot 322307
3276  Utot 305416
3277  Utot 289577
3278  Utot 319865
3279  Utot 291017
3280  Utot 350348
3281  Utot 291117
3282  Utot 319175
3283  Utot 349705
3284  Utot 331130
3285  Utot 354829
3286  Utot 344167
3287  Utot 393198
3288  Utot 316021
3289  Utot 305136
3290  Utot 320995
3291  Utot 306873
3292  Utot 314984
3293  Utot 316933
3294  Utot 325476
3295  Utot 329887
3296  Utot 326550
3297  Utot 322599
3298  Utot 300208
3299  Utot 308289
3300  Utot 282234
3301  Utot 343570
3302  Utot 305630
3303  Utot 317130
3304  Utot 325461
3305  Utot 377730
3306  Utot 331395
3307  Utot 336283
3308  Utot 414984
3309  Utot 309861
3310  Utot 297540
3311  Utot 348386
3312  Utot 318674
3313  Utot 332241
3314  Utot 316892
3315  Utot 282132
3316  Utot 366395
3317  Utot 323449
3318  Utot 311130
3319  Utot 367936
3320  Utot 354595
3321  Utot 334223
3322  Utot 300898
3323  Utot 310964
3324  Utot 346291
3325  Utot 274532
3326  Utot 340132
3327  Utot 305848
3328  Utot 356113
3329  Utot 337983
3330  Utot 320035
3331  Utot 343395
3332  Utot 302998
3333  Utot 358437
3334  Utot 328136
3335  Utot 336285
3336  Utot 329448
3337  Utot 301197
3338  Utot 351606
3339  Utot 347226
3340  Utot 397856
3341  Utot 311941
3342  Utot 392597
3343  Utot 312895
3344  Utot 344611
3345  Utot 336046
3346  Utot 277697
3347  Utot 386395
3348  Utot 351065
3349  Utot 301421
3350  Utot 364125
3351  Utot 372102
3352  Utot 295735
3353  Utot 270507
3354  Utot 359570
3355  Utot 331656
3356  Utot 327643
3357  Utot 356744
3358  Utot 308299
3359  Utot 341044
3360  Utot 345805
3361  Utot 315329
3362  Utot 292270
3363  Utot 313618
3364  Utot 348615
3365  Utot 301818
3366  Utot 316265
3367  Utot 312113
3368  Utot 328124
3369  Utot 319163
3370  Utot 303996
3371  Utot 318729
3372  Utot 332602
3373  Utot 340544
3374  Utot 328439
3375  Utot 318486
3376  Utot 336763
3377  Utot 323781
3378  Utot 299927
3379  Utot 385072
3380  Utot 381405
3381  Utot 340967
3382  Utot 404824
3383  Utot 300418
3384  Utot 321577
3385  Utot 346170
3386  Utot 342747
3387  Utot 318681
3388  Utot 368686
3389  Utot 321765
3390  Utot 304749
3391  Utot 318181
3392  Utot 323024
3393  Utot 290334
3394  Utot 367255
3395  Utot 322077
3396  Utot 333151
3397  Utot 273663
3398  Utot 421488
3399  Utot 281060
3400  Utot 328853
3401  Utot 340083
3402  Utot 323511
3403  Utot 365817
3404  Utot 398614
3405  Utot 313396
3406  Utot 361099
3407  Utot 318130
3408  Utot 308627
3409  Utot 335475
3410  Utot 304611
3411  Utot 336832
3412  Utot 325946
3413  Utot 305419
3414  Utot 339600
3415  Utot 284182
3416  Utot 300845
3417  Utot 317903
3418  Utot 373225
3419  Utot 320429
3420  Utot 335270
3421  Utot 316498
3422  Utot 331187
3423  Utot 342105
3424  Utot 329928
3425  Utot 307180
3426  Utot 291055
3427  Utot 325653
3428  Utot 287982
3429  Utot 329282
3430  Utot 327688
3431  Utot 304087
3432  Utot 394472
3433  Utot 298653
3434  Utot 297499
3435  Utot 381106
3436  Utot 386387
3437  Utot 303618
3438  Utot 361920
3439  Utot 362756
3440  Utot 295479
3441  Utot 337286
3442  Utot 313101
3443  Utot 328132
3444  Utot 314601
3445  Utot 320516
3446  Utot 307849
3447  Utot 317970
3448  Utot 341075
3449  Utot 308716
3450  Utot 327974
3451  Utot 344321
3452  Utot 331941
3453  Utot 375869
3454  Utot 284519
3455  Utot 350947
3456  Utot 325400
3457  Utot 362795
3458  Utot 441758
3459  Utot 319149
3460  Utot 292337
3461  Utot 309723
3462  Utot 307216
3463  Utot 316209
3464  Utot 396946
3465  Utot 397341
3466  Utot 344329
3467  Utot 295809
3468  Utot 330915
3469  Utot 282602
3470  Utot 316676
3471  Utot 394063
3472  Utot 348225
3473  Utot 332458
3474  Utot 357927
3475  Utot 316909
3476  Utot 330347
3477  Utot 357280
3478  Utot 283629
3479  Utot 340119
3480  Utot 360729
3481  Utot 363478
3482  Utot 317632
3483  Utot 339109
3484  Utot 303488
3485  Utot 357606
3486  Utot 286332
3487  Utot 344545
3488  Utot 299782
3489  Utot 354757
3490  Utot 301094
3491  Utot 300398
3492  Utot 348916
3493  Utot 314319
3494  Utot 316382
3495  Utot 348474
3496  Utot 281663
3497  Utot 317618
3498  Utot 307333
3499  Utot 291148
3500  Utot 360243
3501  Utot 343744
3502  Utot 351771
3503  Utot 376844
3504  Utot 348113
3505  Utot 294999
3506  Utot 306084
3507  Utot 349652
3508  Utot 304864
3509  Utot 380790
3510  Utot 306550
3511  Utot 397519
3512  Utot 389084
3513  Utot 358750
3514  Utot 340377
3515  Utot 379207
3516  Utot 276941
3517  Utot 307811
3518  Utot 299774
3519  Utot 328458
3520  Utot 357701
3521  Utot 306406
3522  Utot 321261
3523  Utot 376552
3524  Utot 381034
3525  Utot 363904
3526  Utot 362287
3527  Utot 312884
3528  Utot 322978
3529  Utot 331578
3530  Utot 308347
3531  Utot 316224
3532  Utot 364893
3533  Utot 319497
3534  Utot 380975
3535  Utot 335344
3536  Utot 384121
3537  Utot 305382
3538  Utot 314667
3539  Utot 367686
3540  Utot 384884
3541  Utot 354425
3542  Utot 331229
3543  Utot 291962
3544  Utot 279292
3545  Utot 439673
3546  Utot 369066
3547  Utot 360256
3548  Utot 290205
3549  Utot 297244
3550  Utot 349317
3551  Utot 333662
3552  Utot 310414
3553  Utot 322014
3554  Utot 367284
3555  Utot 311984
3556  Utot 316064
3557  Utot 366493
3558  Utot 322368
3559  Utot 285575
3560  Utot 366119
3561  Utot 363280
3562  Utot 402083
3563  Utot 312222
3564  Utot 354162
3565  Utot 397972
3566  Utot 354867
3567  Utot 388625
3568  Utot 343378
3569  Utot 319130
3570  Utot 336047
3571  Utot 354712
3572  Utot 306112
3573  Utot 338774
3574  Utot 305706
3575  Utot 370417
3576  Utot 293058
3577  Utot 328866
3578  Utot 327854
3579  Utot 317285
3580  Utot 326481
3581  Utot 327477
3582  Utot 358720
3583  Utot 323159
3584  Utot 383356
3585  Utot 318950
3586  Utot 322772
3587  Utot 309559
3588  Utot 314231
3589  Utot 287773
3590  Utot 324311
3591  Utot 391160
3592  Utot 307764
3593  Utot 270344
3594  Utot 376168
3595  Utot 313359
3596  Utot 299941
3597  Utot 298950
3598  Utot 426849
3599  Utot 344402
3600  Utot 310006
3601  Utot 401936
3602  Utot 416056
3603  Utot 301053
3604  Utot 309882
3605  Utot 387324
3606  Utot 310696
3607  Utot 349077
3608  Utot 333642
3609  Utot 387137
3610  Utot 345941
3611  Utot 298167
3612  Utot 317640
3613  Utot 301889
3614  Utot 276735
3615  Utot 354978
3616  Utot 302425
3617  Utot 299753
3618  Utot 345337
3619  Utot 317218
3620  Utot 368374
3621  Utot 305388
3622  Utot 336943
3623  Utot 307347
3624  Utot 331343
3625  Utot 314723
3626  Utot 405595
3627  Utot 292076
3628  Utot 314503
3629  Utot 289285
3630  Utot 345087
3631  Utot 289932
3632  Utot 348282
3633  Utot 349601
3634  Utot 403309
3635  Utot 354012
3636  Utot 318940
3637  Utot 400250
3638  Utot 312030
3639  Utot 311182
3640  Utot 352394
3641  Utot 323492
3642  Utot 287161
3643  Utot 317302
3644  Utot 356013
3645  Utot 308106
3646  Utot 308324
3647  Utot 328946
3648  Utot 327596
3649  Utot 397182
3650  Utot 359423
3651  Utot 327685
3652  Utot 351713
3653  Utot 315934
3654  Utot 302174
3655  Utot 280141
3656  Utot 324642
3657  Utot 316969
3658  Utot 335132
3659  Utot 319213
3660  Utot 309729
3661  Utot 338104
3662  Utot 315484
3663  Utot 369319
3664  Utot 290901
3665  Utot 306892
3666  Utot 350495
3667  Utot 307276
3668  Utot 344992
3669  Utot 314381
3670  Utot 319737
3671  Utot 312299
3672  Utot 351639
3673  Utot 397329
3674  Utot 344548
3675  Utot 343493
3676  Utot 295903
3677  Utot 363160
3678  Utot 341854
3679  Utot 318352
3680  Utot 387203
3681  Utot 338075
3682  Utot 282182
3683  Utot 360347
3684  Utot 311237
3685  Utot 303558
3686  Utot 318263
3687  Utot 342184
3688  Utot 276350
3689  Utot 341011
3690  Utot 288820
3691  Utot 387536
3692  Utot 375639
3693  Utot 364725
3694  Utot 330004
3695  Utot 324741
3696  Utot 312516
3697  Utot 343721
3698  Utot 342832
3699  Utot 283036
3700  Utot 356831
3701  Utot 304061
3702  Utot 337470
3703  Utot 311004
3704  Utot 334143
3705  Utot 325000
3706  Utot 329199
3707  Utot 388949
3708  Utot 282427
3709  Utot 284698
3710  Utot 415569
3711  Utot 322870
3712  Utot 342044
3713  Utot 297457
3714  Utot 299826
3715  Utot 302135
3716  Utot 370174
3717  Utot 323415
3718  Utot 308824
3719  Utot 299304
3720  Utot 360114
3721  Utot 337414
3722  Utot 412503
3723  Utot 289843
3724  Utot 346093
3725  Utot 325007
3726  Utot 319073
3727  Utot 346466
3728  Utot 289645
3729  Utot 330818
3730  Utot 348919
3731  Utot 345312
3732  Utot 361445
3733  Utot 293511
3734  Utot 292391
3735  Utot 355630
3736  Utot 304538
3737  Utot 321460
3738  Utot 389265
3739  Utot 296345
3740  Utot 329217
3741  Utot 318229
3742  Utot 436738
3743  Utot 334571
3744  Utot 330290
3745  Utot 307626
3746  Utot 366360
3747  Utot 319242
3748  Utot 380798
3749  Utot 317050
3750  Utot 372152
3751  Utot 291345
3752  Utot 305340
3753  Utot 342935
3754  Utot 278548
3755  Utot 279276
3756  Utot 405884
3757  Utot 319022
3758  Utot 302212
3759  Utot 378199
3760  Utot 330044
3761  Utot 304143
3762  Utot 320603
3763  Utot 292407
3764  Utot 327446
3765  Utot 339419
3766  Utot 329780
3767  Utot 308169
3768  Utot 431090
3769  Utot 315722
3770  Utot 323660
3771  Utot 296504
3772  Utot 386872
3773  Utot 360666
3774  Utot 355503
3775  Utot 290338
3776  Utot 366955
3777  Utot 294290
3778  Utot 342069
3779  Utot 301790
3780  Utot 409855
3781  Utot 336033
3782  Utot 352517
3783  Utot 362291
3784  Utot 320842
3785  Utot 288604
3786  Utot 297543
3787  Utot 351646
3788  Utot 371899
3789  Utot 305904
3790  Utot 353526
3791  Utot 353725
3792  Utot 320647
3793  Utot 341123
3794  Utot 359162
3795  Utot 335685
3796  Utot 342810
3797  Utot 302210
3798  Utot 372043
3799  Utot 271885
3800  Utot 322565
3801  Utot 363883
3802  Utot 345605
3803  Utot 295823
3804  Utot 293075
3805  Utot 319234
3806  Utot 317716
3807  Utot 311683
3808  Utot 316746
3809  Utot 357654
3810  Utot 293464
3811  Utot 346653
3812  Utot 269999
3813  Utot 337145
3814  Utot 329083
3815  Utot 404227
3816  Utot 300085
3817  Utot 336648
3818  Utot 314665
3819  Utot 315760
3820  Utot 329825
3821  Utot 346975
3822  Utot 340504
3823  Utot 308496
3824  Utot 367959
3825  Utot 343723
3826  Utot 375021
3827  Utot 366360
3828  Utot 328767
3829  Utot 394910
3830  Utot 335291
3831  Utot 289292
3832  Utot 323233
3833  Utot 347084
3834  Utot 307573
3835  Utot 332670
3836  Utot 363532
3837  Utot 330720
3838  Utot 324787
3839  Utot 286971
3840  Utot 356252
3841  Utot 337506
3842  Utot 365952
3843  Utot 404385
3844  Utot 326574
3845  Utot 453872
3846  Utot 313975
3847  Utot 307856
3848  Utot 348445
3849  Utot 330443
3850  Utot 323771
3851  Utot 292452
3852  Utot 352932
3853  Utot 328855
3854  Utot 297890
3855  Utot 372177
3856  Utot 304509
3857  Utot 299530
3858  Utot 305857
3859  Utot 326108
3860  Utot 293858
3861  Utot 464008
3862  Utot 395095
3863  Utot 336031
3864  Utot 426352
3865  Utot 345367
3866  Utot 353040
3867  Utot 302244
3868  Utot 360738
3869  Utot 371086
3870  Utot 285674
3871  Utot 336551
3872  Utot 337915
3873  Utot 310025
3874  Utot 324596
3875  Utot 292009
3876  Utot 356371
3877  Utot 353906
3878  Utot 364683
3879  Utot 340141
3880  Utot 391484
3881  Utot 350498
3882  Utot 397833
3883  Utot 327955
3884  Utot 339472
3885  Utot 367942
3886  Utot 333585
3887  Utot 315881
3888  Utot 300706
3889  Utot 317387
3890  Utot 342830
3891  Utot 313756
3892  Utot 376214
3893  Utot 320242
3894  Utot 305577
3895  Utot 324042
3896  Utot 298757
3897  Utot 287757
3898  Utot 344067
3899  Utot 371772
3900  Utot 354151
3901  Utot 320112
3902  Utot 331728
3903  Utot 363959
3904  Utot 346054
3905  Utot 315971
3906  Utot 377354
3907  Utot 278391
3908  Utot 331453
3909  Utot 348334
3910  Utot 340336
3911  Utot 273509
3912  Utot 301418
3913  Utot 321620
3914  Utot 340456
3915  Utot 355926
3916  Utot 370301
3917  Utot 296202
3918  Utot 328190
3919  Utot 350949
3920  Utot 306297
3921  Utot 293520
3922  Utot 322917
3923  Utot 303168
3924  Utot 388624
3925  Utot 350028
3926  Utot 302126
3927  Utot 338866
3928  Utot 301575
3929  Utot 350116
3930  Utot 314688
3931  Utot 366622
3932  Utot 287485
3933  Utot 275805
3934  Utot 280636
3935  Utot 279532
3936  Utot 303832
3937  Utot 408114
3938  Utot 319999
3939  Utot 306067
3940  Utot 286646
3941  Utot 343150
3942  Utot 327889
3943  Utot 316710
3944  Utot 309393
3945  Utot 300881
3946  Utot 321724
3947  Utot 392933
3948  Utot 318407
3949  Utot 388614
3950  Utot 292722
3951  Utot 370785
3952  Utot 364124
3953  Utot 304116
3954  Utot 341516
3955  Utot 319390
3956  Utot 427070
3957  Utot 303459
3958  Utot 317078
3959  Utot 293655
3960  Utot 306368
3961  Utot 327389
3962  Utot 420568
3963  Utot 368079
3964  Utot 301262
3965  Utot 368563
3966  Utot 396256
3967  Utot 348312
3968  Utot 294767
3969  Utot 315575
3970  Utot 333494
3971  Utot 295763
3972  Utot 321683
3973  Utot 283401
3974  Utot 332883
3975  Utot 338721
3976  Utot 287182
3977  Utot 388965
3978  Utot 295082
3979  Utot 286024
3980  Utot 343088
3981  Utot 278615
3982  Utot 320835
3983  Utot 369944
3984  Utot 360239
3985  Utot 295208
3986  Utot 344625
3987  Utot 359397
3988  Utot 338675
3989  Utot 287861
3990  Utot 330360
3991  Utot 343033
3992  Utot 335524
3993  Utot 303065
3994  Utot 319458
3995  Utot 303037
3996  Utot 364017
3997  Utot 282952
3998  Utot 289987
3999  Utot 356095
4000  Utot 320999
4001  Utot 309517
4002  Utot 312588
4003  Utot 322867
4004  Utot 327796
4005  Utot 326815
4006  Utot 322541
4007  Utot 306319
4008  Utot 354241
4009  Utot 362107
4010  Utot 324709
4011  Utot 318138
4012  Utot 320788
4013  Utot 308461
4014  Utot 288560
4015  Utot 292402
4016  Utot 309415
4017  Utot 368988
4018  Utot 350851
4019  Utot 340069
4020  Utot 269893
4021  Utot 334923
4022  Utot 325517
4023  Utot 309505
4024  Utot 301407
4025  Utot 362042
4026  Utot 321308
4027  Utot 366375
4028  Utot 309831
4029  Utot 295085
4030  Utot 348947
4031  Utot 326073
4032  Utot 313725
4033  Utot 296369
4034  Utot 368794
4035  Utot 402292
4036  Utot 311035
4037  Utot 380131
4038  Utot 332482
4039  Utot 287965
4040  Utot 349126
4041  Utot 363205
4042  Utot 294552
4043  Utot 342943
4044  Utot 382680
4045  Utot 350504
4046  Utot 322236
4047  Utot 312312
4048  Utot 370549
4049  Utot 330637
4050  Utot 332342
4051  Utot 333639
4052  Utot 351810
4053  Utot 336903
4054  Utot 339000
4055  Utot 308495
4056  Utot 320248
4057  Utot 396263
4058  Utot 302372
4059  Utot 286963
4060  Utot 328195
4061  Utot 350295
4062  Utot 294354
4063  Utot 435149
4064  Utot 345277
4065  Utot 315508
4066  Utot 274793
4067  Utot 286483
4068  Utot 329077
4069  Utot 356805
4070  Utot 397268
4071  Utot 355933
4072  Utot 323752
4073  Utot 326613
4074  Utot 430761
4075  Utot 303874
4076  Utot 336579
4077  Utot 288161
4078  Utot 381308
4079  Utot 386321
4080  Utot 331884
4081  Utot 314846
4082  Utot 299728
4083  Utot 337055
4084  Utot 340944
4085  Utot 361206
4086  Utot 342136
4087  Utot 338354
4088  Utot 300555
4089  Utot 330982
4090  Utot 305461
4091  Utot 284286
4092  Utot 351729
4093  Utot 410143
4094  Utot 343282
4095  Utot 339348
4096  Utot 291482
4097  Utot 311604
4098  Utot 328764
4099  Utot 326003
4100  Utot 322338
4101  Utot 300163
4102  Utot 314485
4103  Utot 358970
4104  Utot 331451
4105  Utot 397346
4106  Utot 322603
4107  Utot 295226
4108  Utot 274526
4109  Utot 278609
4110  Utot 321958
4111  Utot 411542
4112  Utot 312420
4113  Utot 320502
4114  Utot 279053
4115  Utot 280480
4116  Utot 317143
4117  Utot 335443
4118  Utot 304483
4119  Utot 299670
4120  Utot 391082
4121  Utot 322888
4122  Utot 343419
4123  Utot 335820
4124  Utot 337171
4125  Utot 357039
4126  Utot 293349
4127  Utot 322200
4128  Utot 321135
4129  Utot 329483
4130  Utot 320537
4131  Utot 278861
4132  Utot 296395
4133  Utot 371891
4134  Utot 306420
4135  Utot 300436
4136  Utot 304609
4137  Utot 381946
4138  Utot 300274
4139  Utot 371335
4140  Utot 345040
4141  Utot 310855
4142  Utot 299477
4143  Utot 271283
4144  Utot 324254
4145  Utot 339245
4146  Utot 283502
4147  Utot 378853
4148  Utot 388392
4149  Utot 308234
4150  Utot 285566
4151  Utot 330996
4152  Utot 349263
4153  Utot 350320
4154  Utot 343619
4155  Utot 326563
4156  Utot 317641
4157  Utot 334619
4158  Utot 328537
4159  Utot 321407
4160  Utot 283601
4161  Utot 350679
4162  Utot 336616
4163  Utot 315801
4164  Utot 299433
4165  Utot 288513
4166  Utot 350246
4167  Utot 396992
4168  Utot 287576
4169  Utot 369461
4170  Utot 315622
4171  Utot 297742
4172  Utot 286255
4173  Utot 335053
4174  Utot 351558
4175  Utot 316650
4176  Utot 390424
4177  Utot 367441
4178  Utot 341444
4179  Utot 292415
4180  Utot 322791
4181  Utot 355850
4182  Utot 381713
4183  Utot 291271
4184  Utot 317856
4185  Utot 349888
4186  Utot 366977
4187  Utot 393226
4188  Utot 333661
4189  Utot 312977
4190  Utot 333271
4191  Utot 334336
4192  Utot 299047
4193  Utot 319598
4194  Utot 309240
4195  Utot 326695
4196  Utot 300915
4197  Utot 326107
4198  Utot 295911
4199  Utot 316146
4200  Utot 392277
4201  Utot 347889
4202  Utot 293556
4203  Utot 349063
4204  Utot 343929
4205  Utot 383196
4206  Utot 346945
4207  Utot 365138
4208  Utot 365112
4209  Utot 325862
4210  Utot 305392
4211  Utot 300356
4212  Utot 294615
4213  Utot 360131
4214  Utot 346160
4215  Utot 300941
4216  Utot 349893
4217  Utot 318093
4218  Utot 408603
4219  Utot 377774
4220  Utot 381281
4221  Utot 286414
4222  Utot 292646
4223  Utot 322696
4224  Utot 295908
4225  Utot 309068
4226  Utot 277827
4227  Utot 338986
4228  Utot 336339
4229  Utot 396936
4230  Utot 396568
4231  Utot 345665
4232  Utot 317698
4233  Utot 314867
4234  Utot 298715
4235  Utot 319058
4236  Utot 345928
4237  Utot 357468
4238  Utot 312411
4239  Utot 300240
4240  Utot 398050
4241  Utot 353652
4242  Utot 343973
4243  Utot 336476
4244  Utot 337267
4245  Utot 295404
4246  Utot 319513
4247  Utot 292226
4248  Utot 339883
4249  Utot 314618
4250  Utot 336749
4251  Utot 348327
4252  Utot 283332
4253  Utot 337294
4254  Utot 304721
4255  Utot 342657
4256  Utot 314895
4257  Utot 380218
4258  Utot 323504
4259  Utot 301185
4260  Utot 362877
4261  Utot 299461
4262  Utot 322918
4263  Utot 304963
4264  Utot 289052
4265  Utot 317534
4266  Utot 391769
4267  Utot 347494
4268  Utot 369647
4269  Utot 348785
4270  Utot 369352
4271  Utot 338301
4272  Utot 312025
4273  Utot 347791
4274  Utot 341127
4275  Utot 338236
4276  Utot 316126
4277  Utot 300982
4278  Utot 310734
4279  Utot 351281
4280  Utot 369082
4281  Utot 310314
4282  Utot 317992
4283  Utot 310137
4284  Utot 322275
4285  Utot 339982
4286  Utot 400419
4287  Utot 354064
4288  Utot 288211
4289  Utot 376047
4290  Utot 326226
4291  Utot 325894
4292  Utot 327255
4293  Utot 333460
4294  Utot 406600
4295  Utot 343837
4296  Utot 364909
4297  Utot 323076
4298  Utot 409630
4299  Utot 290006
4300  Utot 335972
4301  Utot 407474
4302  Utot 305847
4303  Utot 291940
4304  Utot 335235
4305  Utot 359965
4306  Utot 354282
4307  Utot 341303
4308  Utot 340496
4309  Utot 409582
4310  Utot 313966
4311  Utot 327435
4312  Utot 335516
4313  Utot 270392
4314  Utot 365573
4315  Utot 356642
4316  Utot 435457
4317  Utot 363238
4318  Utot 299934
4319  Utot 314334
4320  Utot 334817
4321  Utot 370880
4322  Utot 376260
4323  Utot 375416
4324  Utot 288419
4325  Utot 297369
4326  Utot 325253
4327  Utot 307522
4328  Utot 333760
4329  Utot 357080
4330  Utot 363226
4331  Utot 331668
4332  Utot 291732
4333  Utot 318841
4334  Utot 301969
4335  Utot 316528
4336  Utot 321979
4337  Utot 330857
4338  Utot 275680
4339  Utot 334096
4340  Utot 421074
4341  Utot 361450
4342  Utot 299653
4343  Utot 306351
4344  Utot 384808
4345  Utot 491686
4346  Utot 315537
4347  Utot 374372
4348  Utot 340073
4349  Utot 346555
4350  Utot 279052
4351  Utot 349494
4352  Utot 303019
4353  Utot 387574
4354  Utot 294883
4355  Utot 312373
4356  Utot 353484
4357  Utot 389592
4358  Utot 293070
4359  Utot 415813
4360  Utot 290618
4361  Utot 320722
4362  Utot 283334
4363  Utot 304530
4364  Utot 310048
4365  Utot 324066
4366  Utot 304019
4367  Utot 275250
4368  Utot 300872
4369  Utot 373382
4370  Utot 302083
4371  Utot 374300
4372  Utot 270885
4373  Utot 330325
4374  Utot 325703
4375  Utot 389357
4376  Utot 297233
4377  Utot 368817
4378  Utot 331733
4379  Utot 367418
4380  Utot 327603
4381  Utot 348717
4382  Utot 311821
4383  Utot 363904
4384  Utot 383259
4385  Utot 372151
4386  Utot 328548
4387  Utot 304069
4388  Utot 285561
4389  Utot 287325
4390  Utot 324129
4391  Utot 317587
4392  Utot 296902
4393  Utot 299314
4394  Utot 414337
4395  Utot 345175
4396  Utot 333388
4397  Utot 307417
4398  Utot 276554
4399  Utot 324348
4400  Utot 304017
4401  Utot 291465
4402  Utot 307111
4403  Utot 327246
4404  Utot 301868
4405  Utot 333470
4406  Utot 295765
4407  Utot 353233
4408  Utot 395190
4409  Utot 303389
4410  Utot 359349
4411  Utot 374163
4412  Utot 337989
4413  Utot 306283
4414  Utot 309859
4415  Utot 303242
4416  Utot 305291
4417  Utot 297805
4418  Utot 306291
4419  Utot 323168
4420  Utot 370314
4421  Utot 277379
4422  Utot 339437
4423  Utot 303457
4424  Utot 327791
4425  Utot 384897
4426  Utot 301780
4427  Utot 358419
4428  Utot 283578
4429  Utot 347153
4430  Utot 329124
4431  Utot 347776
4432  Utot 323384
4433  Utot 288983
4434  Utot 302476
4435  Utot 335813
4436  Utot 290968
4437  Utot 296752
4438  Utot 319034
4439  Utot 316595
4440  Utot 324994
4441  Utot 334581
4442  Utot 341005
4443  Utot 320431
4444  Utot 376555
4445  Utot 289089
4446  Utot 382144
4447  Utot 301044
4448  Utot 377419
4449  Utot 293335
4450  Utot 363796
4451  Utot 307394
4452  Utot 297006
4453  Utot 297833
4454  Utot 277465
4455  Utot 287507
4456  Utot 351841
4457  Utot 310124
4458  Utot 285655
4459  Utot 287526
4460  Utot 346665
4461  Utot 296193
4462  Utot 336076
4463  Utot 327852
4464  Utot 310775
4465  Utot 295341
4466  Utot 363939
4467  Utot 338397
4468  Utot 357301
4469  Utot 312577
4470  Utot 321795
4471  Utot 314022
4472  Utot 301699
4473  Utot 351810
4474  Utot 321051
4475  Utot 404670
4476  Utot 433685
4477  Utot 277913
4478  Utot 317668
4479  Utot 394612
4480  Utot 335706
4481  Utot 351673
4482  Utot 365959
4483  Utot 375599
4484  Utot 385994
4485  Utot 317233
4486  Utot 305485
4487  Utot 300754
4488  Utot 332398
4489  Utot 334354
4490  Utot 385966
4491  Utot 300512
4492  Utot 336653
4493  Utot 307449
4494  Utot 327144
4495  Utot 342651
4496  Utot 335039
4497  Utot 357227
4498  Utot 297840
4499  Utot 387801
4500  Utot 332766
4501  Utot 369527
4502  Utot 323197
4503  Utot 362512
4504  Utot 325044
4505  Utot 350146
4506  Utot 310703
4507  Utot 310814
4508  Utot 326773
4509  Utot 388196
4510  Utot 317693
4511  Utot 311118
4512  Utot 355448
4513  Utot 325407
4514  Utot 396008
4515  Utot 321466
4516  Utot 320678
4517  Utot 311818
4518  Utot 285139
4519  Utot 269387
4520  Utot 352420
4521  Utot 360104
4522  Utot 349329
4523  Utot 299509
4524  Utot 313606
4525  Utot 313568
4526  Utot 345842
4527  Utot 341890
4528  Utot 291662
4529  Utot 297651
4530  Utot 342872
4531  Utot 308667
4532  Utot 311685
4533  Utot 431737
4534  Utot 298892
4535  Utot 338175
4536  Utot 337385
4537  Utot 349333
4538  Utot 337186
4539  Utot 376022
4540  Utot 296684
4541  Utot 337398
4542  Utot 307413
4543  Utot 340376
4544  Utot 325000
4545  Utot 325609
4546  Utot 328427
4547  Utot 310318
4548  Utot 304150
4549  Utot 353804
4550  Utot 354522
4551  Utot 318884
4552  Utot 408743
4553  Utot 337006
4554  Utot 340480
4555  Utot 354406
4556  Utot 354136
4557  Utot 329138
4558  Utot 352090
4559  Utot 321557
4560  Utot 399768
4561  Utot 349533
4562  Utot 352613
4563  Utot 313137
4564  Utot 288987
4565  Utot 287730
4566  Utot 324080
4567  Utot 358246
4568  Utot 332403
4569  Utot 348645
4570  Utot 309434
4571  Utot 298403
4572  Utot 313370
4573  Utot 339776
4574  Utot 323996
4575  Utot 311472
4576  Utot 296726
4577  Utot 398287
4578  Utot 350900
4579  Utot 317040
4580  Utot 366209
4581  Utot 272486
4582  Utot 359157
4583  Utot 314301
4584  Utot 353349
4585  Utot 340361
4586  Utot 288147
4587  Utot 320938
4588  Utot 330807
4589  Utot 303047
4590  Utot 309295
4591  Utot 281967
4592  Utot 315254
4593  Utot 311340
4594  Utot 310221
4595  Utot 305451
4596  Utot 368466
4597  Utot 309387
4598  Utot 313834
4599  Utot 334359
4600  Utot 482422
4601  Utot 418944
4602  Utot 378933
4603  Utot 379659
4604  Utot 350595
4605  Utot 288694
4606  Utot 347653
4607  Utot 374596
4608  Utot 362381
4609  Utot 308415
4610  Utot 308078
4611  Utot 312334
4612  Utot 439375
4613  Utot 309252
4614  Utot 276795
4615  Utot 342188
4616  Utot 385855
4617  Utot 350628
4618  Utot 307932
4619  Utot 429431
4620  Utot 346385
4621  Utot 343478
4622  Utot 338645
4623  Utot 386522
4624  Utot 287049
4625  Utot 371993
4626  Utot 344964
4627  Utot 326475
4628  Utot 283019
4629  Utot 309148
4630  Utot 350678
4631  Utot 362891
4632  Utot 362228
4633  Utot 297245
4634  Utot 403649
4635  Utot 351908
4636  Utot 306090
4637  Utot 330758
4638  Utot 321602
4639  Utot 412554
4640  Utot 310207
4641  Utot 321932
4642  Utot 304624
4643  Utot 304235
4644  Utot 303118
4645  Utot 277448
4646  Utot 358474
4647  Utot 325141
4648  Utot 288284
4649  Utot 341697
4650  Utot 333747
4651  Utot 347359
4652  Utot 341079
4653  Utot 321479
4654  Utot 305714
4655  Utot 326165
4656  Utot 355457
4657  Utot 348118
4658  Utot 342933
4659  Utot 304029
4660  Utot 322996
4661  Utot 344500
4662  Utot 288004
4663  Utot 360069
4664  Utot 328502
4665  Utot 349753
4666  Utot 294333
4667  Utot 341209
4668  Utot 306004
4669  Utot 310616
4670  Utot 273328
4671  Utot 377467
4672  Utot 312733
4673  Utot 356758
4674  Utot 297048
4675  Utot 293767
4676  Utot 331749
4677  Utot 303497
4678  Utot 319438
4679  Utot 314840
4680  Utot 323942
4681  Utot 303943
4682  Utot 309156
4683  Utot 395225
4684  Utot 318271
4685  Utot 341406
4686  Utot 323499
4687  Utot 355577
4688  Utot 284246
4689  Utot 318004
4690  Utot 531650
4691  Utot 317232
4692  Utot 330485
4693  Utot 365240
4694  Utot 317173
4695  Utot 317247
4696  Utot 301197
4697  Utot 319992
4698  Utot 310669
4699  Utot 353616
4700  Utot 286057
4701  Utot 333180
4702  Utot 354658
4703  Utot 350569
4704  Utot 399930
4705  Utot 425114
4706  Utot 350893
4707  Utot 326675
4708  Utot 291227
4709  Utot 302924
4710  Utot 288660
4711  Utot 314038
4712  Utot 355219
4713  Utot 329935
4714  Utot 298501
4715  Utot 333282
4716  Utot 289096
4717  Utot 335544
4718  Utot 364563
4719  Utot 331398
4720  Utot 387442
4721  Utot 329339
4722  Utot 308977
4723  Utot 331906
4724  Utot 344165
4725  Utot 344302
4726  Utot 258212
4727  Utot 305643
4728  Utot 301224
4729  Utot 286710
4730  Utot 351040
4731  Utot 311159
4732  Utot 322179
4733  Utot 267184
4734  Utot 312279
4735  Utot 358488
4736  Utot 277816
4737  Utot 321982
4738  Utot 338329
4739  Utot 326450
4740  Utot 356450
4741  Utot 323313
4742  Utot 298795
4743  Utot 333257
4744  Utot 372440
4745  Utot 343593
4746  Utot 308027
4747  Utot 379580
4748  Utot 301414
4749  Utot 379186
4750  Utot 380246
4751  Utot 379891
4752  Utot 349734
4753  Utot 335523
4754  Utot 360059
4755  Utot 316868
4756  Utot 284420
4757  Utot 397729
4758  Utot 347053
4759  Utot 330636
4760  Utot 383314
4761  Utot 339826
4762  Utot 320930
4763  Utot 321955
4764  Utot 327738
4765  Utot 332831
4766  Utot 289712
4767  Utot 340964
4768  Utot 312505
4769  Utot 355681
4770  Utot 380971
4771  Utot 342538
4772  Utot 329096
4773  Utot 315439
4774  Utot 335231
4775  Utot 261982
4776  Utot 321523
4777  Utot 372489
4778  Utot 290427
4779  Utot 343557
4780  Utot 286171
4781  Utot 370594
4782  Utot 313610
4783  Utot 303898
4784  Utot 319235
4785  Utot 378689
4786  Utot 312540
4787  Utot 369341
4788  Utot 311874
4789  Utot 303199
4790  Utot 309943
4791  Utot 387636
4792  Utot 368033
4793  Utot 318252
4794  Utot 316045
4795  Utot 348378
4796  Utot 353164
4797  Utot 294163
4798  Utot 335978
4799  Utot 316035
4800  Utot 276874
4801  Utot 339500
4802  Utot 298708
4803  Utot 305347
4804  Utot 356114
4805  Utot 320397
4806  Utot 359144
4807  Utot 325949
4808  Utot 328058
4809  Utot 313206
4810  Utot 334392
4811  Utot 306712
4812  Utot 328946
4813  Utot 422009
4814  Utot 341855
4815  Utot 322058
4816  Utot 310476
4817  Utot 346806
4818  Utot 339557
4819  Utot 359327
4820  Utot 371199
4821  Utot 352317
4822  Utot 268555
4823  Utot 360213
4824  Utot 271691
4825  Utot 323097
4826  Utot 333862
4827  Utot 328684
4828  Utot 337065
4829  Utot 317697
4830  Utot 279510
4831  Utot 345535
4832  Utot 381042
4833  Utot 381288
4834  Utot 324548
4835  Utot 309344
4836  Utot 336196
4837  Utot 316445
4838  Utot 415049
4839  Utot 267743
4840  Utot 310987
4841  Utot 329454
4842  Utot 332965
4843  Utot 321883
4844  Utot 287800
4845  Utot 297456
4846  Utot 305771
4847  Utot 339178
4848  Utot 331808
4849  Utot 309308
4850  Utot 309394
4851  Utot 303350
4852  Utot 352351
4853  Utot 384560
4854  Utot 332504
4855  Utot 291269
4856  Utot 327207
4857  Utot 350007
4858  Utot 322728
4859  Utot 321677
4860  Utot 298849
4861  Utot 283178
4862  Utot 339045
4863  Utot 300002
4864  Utot 290841
4865  Utot 344667
4866  Utot 350846
4867  Utot 314446
4868  Utot 326935
4869  Utot 365892
4870  Utot 293008
4871  Utot 327843
4872  Utot 367301
4873  Utot 354863
4874  Utot 414318
4875  Utot 312339
4876  Utot 336358
4877  Utot 323797
4878  Utot 314656
4879  Utot 418745
4880  Utot 278793
4881  Utot 348513
4882  Utot 344558
4883  Utot 308421
4884  Utot 330632
4885  Utot 303139
4886  Utot 333747
4887  Utot 308333
4888  Utot 316962
4889  Utot 370712
4890  Utot 307243
4891  Utot 373695
4892  Utot 296909
4893  Utot 287272
4894  Utot 311104
4895  Utot 307523
4896  Utot 316638
4897  Utot 288836
4898  Utot 303706
4899  Utot 295231
4900  Utot 317389
4901  Utot 344049
4902  Utot 354845
4903  Utot 330890
4904  Utot 344689
4905  Utot 430064
4906  Utot 321525
4907  Utot 298745
4908  Utot 331508
4909  Utot 312687
4910  Utot 315019
4911  Utot 307334
4912  Utot 282819
4913  Utot 374261
4914  Utot 317247
4915  Utot 308109
4916  Utot 402934
4917  Utot 315908
4918  Utot 404126
4919  Utot 307858
4920  Utot 324004
4921  Utot 323057
4922  Utot 352573
4923  Utot 338636
4924  Utot 398861
4925  Utot 295132
4926  Utot 317696
4927  Utot 319995
4928  Utot 403520
4929  Utot 385366
4930  Utot 313225
4931  Utot 342600
4932  Utot 346908
4933  Utot 299258
4934  Utot 369342
4935  Utot 285161
4936  Utot 379037
4937  Utot 286158
4938  Utot 382145
4939  Utot 299881
4940  Utot 283821
4941  Utot 315964
4942  Utot 334896
4943  Utot 301990
4944  Utot 391523
4945  Utot 366070
4946  Utot 293920
4947  Utot 310426
4948  Utot 344152
4949  Utot 284610
4950  Utot 351674
4951  Utot 357216
4952  Utot 280719
4953  Utot 294114
4954  Utot 314118
4955  Utot 286721
4956  Utot 334686
4957  Utot 309443
4958  Utot 358198
4959  Utot 294786
4960  Utot 302961
4961  Utot 310301
4962  Utot 299733
4963  Utot 304978
4964  Utot 307943
4965  Utot 337864
4966  Utot 332473
4967  Utot 286807
4968  Utot 395050
4969  Utot 379443
4970  Utot 307047
4971  Utot 336115
4972  Utot 310569
4973  Utot 403848
4974  Utot 437853
4975  Utot 329901
4976  Utot 301386
4977  Utot 396975
4978  Utot 436951
4979  Utot 292054
4980  Utot 352964
4981  Utot 375196
4982  Utot 334095
4983  Utot 359041
4984  Utot 308853
4985  Utot 282272
4986  Utot 298715
4987  Utot 318503
4988  Utot 312953
4989  Utot 306595
4990  Utot 321196
4991  Utot 344695
4992  Utot 336009
4993  Utot 339073
4994  Utot 324348
4995  Utot 377031
4996  Utot 398937
4997  Utot 318986
4998  Utot 325407
4999  Utot 317193
5000  Utot 318075
5001  Utot 418278
5002  Utot 321835
5003  Utot 384411
5004  Utot 345399
5005  Utot 288506
5006  Utot 302236
5007  Utot 317342
5008  Utot 371791
5009  Utot 281966
5010  Utot 325114
5011  Utot 352674
5012  Utot 306917
5013  Utot 344602
5014  Utot 316220
5015  Utot 323170
5016  Utot 308837
5017  Utot 306924
5018  Utot 341248
5019  Utot 335467
5020  Utot 308891
5021  Utot 321200
5022  Utot 369773
5023  Utot 309685
5024  Utot 317177
5025  Utot 311311
5026  Utot 287324
5027  Utot 290952
5028  Utot 348872
5029  Utot 321895
5030  Utot 323675
5031  Utot 323782
5032  Utot 353224
5033  Utot 321763
5034  Utot 282365
5035  Utot 385522
5036  Utot 307112
5037  Utot 326512
5038  Utot 346617
5039  Utot 295820
5040  Utot 364695
5041  Utot 369167
5042  Utot 333924
5043  Utot 450902
5044  Utot 346127
5045  Utot 309516
5046  Utot 373196
5047  Utot 326770
5048  Utot 385171
5049  Utot 294645
5050  Utot 356133
5051  Utot 308485
5052  Utot 343639
5053  Utot 344448
5054  Utot 373497
5055  Utot 321190
5056  Utot 318184
5057  Utot 308966
5058  Utot 292944
5059  Utot 359232
5060  Utot 326602
5061  Utot 293236
5062  Utot 308197
5063  Utot 321879
5064  Utot 286950
5065  Utot 314108
5066  Utot 341138
5067  Utot 335342
5068  Utot 342348
5069  Utot 374332
5070  Utot 356779
5071  Utot 301277
5072  Utot 344100
5073  Utot 338235
5074  Utot 359921
5075  Utot 305569
5076  Utot 305802
5077  Utot 313366
5078  Utot 372913
5079  Utot 309933
5080  Utot 395100
5081  Utot 357751
5082  Utot 308668
5083  Utot 362357
5084  Utot 293481
5085  Utot 333654
5086  Utot 344688
5087  Utot 364036
5088  Utot 330493
5089  Utot 337524
5090  Utot 315847
5091  Utot 309933
5092  Utot 358409
5093  Utot 301539
5094  Utot 332751
5095  Utot 384509
5096  Utot 311679
5097  Utot 320981
5098  Utot 333004
5099  Utot 378325
5100  Utot 331539
5101  Utot 327473
5102  Utot 386770
5103  Utot 314071
5104  Utot 308874
5105  Utot 315400
5106  Utot 312991
5107  Utot 286404
5108  Utot 322394
5109  Utot 346367
5110  Utot 305535
5111  Utot 334286
5112  Utot 345774
5113  Utot 329300
5114  Utot 368850
5115  Utot 364635
5116  Utot 346538
5117  Utot 333814
5118  Utot 322053
5119  Utot 327054
5120  Utot 337500
5121  Utot 341079
5122  Utot 326913
5123  Utot 356613
5124  Utot 316037
5125  Utot 346060
5126  Utot 383330
5127  Utot 334802
5128  Utot 332979
5129  Utot 331271
5130  Utot 333577
5131  Utot 402626
5132  Utot 304966
5133  Utot 360070
5134  Utot 344765
5135  Utot 289338
5136  Utot 332089
5137  Utot 346266
5138  Utot 321094
5139  Utot 399873
5140  Utot 341469
5141  Utot 315468
5142  Utot 352643
5143  Utot 344279
5144  Utot 339729
5145  Utot 323373
5146  Utot 308313
5147  Utot 356553
5148  Utot 318828
5149  Utot 333987
5150  Utot 310419
5151  Utot 277552
5152  Utot 399765
5153  Utot 369233
5154  Utot 347190
5155  Utot 327696
5156  Utot 286261
5157  Utot 291816
5158  Utot 334136
5159  Utot 346861
5160  Utot 319431
5161  Utot 303031
5162  Utot 329030
5163  Utot 328944
5164  Utot 335289
5165  Utot 336891
5166  Utot 395786
5167  Utot 282069
5168  Utot 322783
5169  Utot 332505
5170  Utot 317464
5171  Utot 321250
5172  Utot 348705
5173  Utot 435947
5174  Utot 376064
5175  Utot 298264
5176  Utot 325676
5177  Utot 338080
5178  Utot 289664
5179  Utot 286709
5180  Utot 318181
5181  Utot 328802
5182  Utot 308117
5183  Utot 321113
5184  Utot 376627
5185  Utot 389641
5186  Utot 327017
5187  Utot 312371
5188  Utot 364682
5189  Utot 316671
5190  Utot 306189
5191  Utot 387585
5192  Utot 300069
5193  Utot 296067
5194  Utot 410599
5195  Utot 281821
5196  Utot 390608
5197  Utot 342460
5198  Utot 292222
5199  Utot 329734
5200  Utot 365211
5201  Utot 317017
5202  Utot 361680
5203  Utot 323858
5204  Utot 314270
5205  Utot 363862
5206  Utot 337296
5207  Utot 343841
5208  Utot 358313
5209  Utot 378601
5210  Utot 313615
5211  Utot 371900
5212  Utot 279738
5213  Utot 363926
5214  Utot 290911
5215  Utot 302490
5216  Utot 340347
5217  Utot 292592
5218  Utot 332754
5219  Utot 356021
5220  Utot 346820
5221  Utot 357260
5222  Utot 314799
5223  Utot 334994
5224  Utot 296166
5225  Utot 288254
5226  Utot 352149
5227  Utot 294737
5228  Utot 354111
5229  Utot 315327
5230  Utot 352583
5231  Utot 315666
5232  Utot 270157
5233  Utot 347424
5234  Utot 361377
5235  Utot 348978
5236  Utot 327523
5237  Utot 291389
5238  Utot 449914
5239  Utot 298318
5240  Utot 377130
5241  Utot 300944
5242  Utot 373219
5243  Utot 305820
5244  Utot 413906
5245  Utot 358422
5246  Utot 344342
5247  Utot 396856
5248  Utot 305597
5249  Utot 287569
5250  Utot 313168
5251  Utot 364150
5252  Utot 324875
5253  Utot 332511
5254  Utot 358274
5255  Utot 303223
5256  Utot 321540
5257  Utot 343556
5258  Utot 332200
5259  Utot 317731
5260  Utot 331689
5261  Utot 368587
5262  Utot 340016
5263  Utot 340734
5264  Utot 323291
5265  Utot 337392
5266  Utot 331440
5267  Utot 402207
5268  Utot 320937
5269  Utot 296423
5270  Utot 323344
5271  Utot 313740
5272  Utot 305001
5273  Utot 462122
5274  Utot 355325
5275  Utot 310943
5276  Utot 369497
5277  Utot 339768
5278  Utot 338059
5279  Utot 354432
5280  Utot 302141
5281  Utot 380144
5282  Utot 288312
5283  Utot 329561
5284  Utot 343160
5285  Utot 316246
5286  Utot 319155
5287  Utot 339396
5288  Utot 340651
5289  Utot 374375
5290  Utot 282548
5291  Utot 319215
5292  Utot 314533
5293  Utot 305876
5294  Utot 319906
5295  Utot 295471
5296  Utot 328159
5297  Utot 290373
5298  Utot 315016
5299  Utot 323482
5300  Utot 392769
5301  Utot 355292
5302  Utot 342562
5303  Utot 349577
5304  Utot 389880
5305  Utot 306389
5306  Utot 360326
5307  Utot 349145
5308  Utot 329382
5309  Utot 314604
5310  Utot 309856
5311  Utot 346602
5312  Utot 282758
5313  Utot 357365
5314  Utot 311388
5315  Utot 345578
5316  Utot 314976
5317  Utot 317666
5318  Utot 329926
5319  Utot 308119
5320  Utot 305496
5321  Utot 280264
5322  Utot 319833
5323  Utot 357976
5324  Utot 348992
5325  Utot 286507
5326  Utot 296203
5327  Utot 329073
5328  Utot 278490
5329  Utot 299028
5330  Utot 298615
5331  Utot 376300
5332  Utot 324887
5333  Utot 315277
5334  Utot 402647
5335  Utot 407593
5336  Utot 384344
5337  Utot 289502
5338  Utot 430228
5339  Utot 333226
5340  Utot 302663
5341  Utot 365908
5342  Utot 345180
5343  Utot 296196
5344  Utot 311621
5345  Utot 312958
5346  Utot 412837
5347  Utot 353352
5348  Utot 368363
5349  Utot 348131
5350  Utot 333134
5351  Utot 296215
5352  Utot 397040
5353  Utot 335967
5354  Utot 306587
5355  Utot 292307
5356  Utot 295353
5357  Utot 270676
5358  Utot 334962
5359  Utot 350739
5360  Utot 324201
5361  Utot 351247
5362  Utot 313407
5363  Utot 347992
5364  Utot 338100
5365  Utot 282119
5366  Utot 346279
5367  Utot 358802
5368  Utot 421002
5369  Utot 327568
5370  Utot 289578
5371  Utot 313569
5372  Utot 378779
5373  Utot 335684
5374  Utot 343737
5375  Utot 346099
5376  Utot 332426
5377  Utot 298826
5378  Utot 340977
5379  Utot 294449
5380  Utot 323671
5381  Utot 278516
5382  Utot 292009
5383  Utot 358089
5384  Utot 328832
5385  Utot 305061
5386  Utot 349175
5387  Utot 356162
5388  Utot 349814
5389  Utot 392602
5390  Utot 315131
5391  Utot 311193
5392  Utot 335819
5393  Utot 297489
5394  Utot 309576
5395  Utot 306905
5396  Utot 279951
5397  Utot 332199
5398  Utot 334383
5399  Utot 336345
5400  Utot 391864
5401  Utot 301331
5402  Utot 306812
5403  Utot 381924
5404  Utot 350117
5405  Utot 291659
5406  Utot 335862
5407  Utot 321476
5408  Utot 362278
5409  Utot 307770
5410  Utot 303412
5411  Utot 298910
5412  Utot 375713
5413  Utot 356109
5414  Utot 289565
5415  Utot 407886
5416  Utot 309671
5417  Utot 322054
5418  Utot 297695
5419  Utot 338942
5420  Utot 289700
5421  Utot 283957
5422  Utot 310127
5423  Utot 337091
5424  Utot 421079
5425  Utot 314154
5426  Utot 320700
5427  Utot 372045
5428  Utot 324459
5429  Utot 367452
5430  Utot 269663
5431  Utot 299264
5432  Utot 282425
5433  Utot 315710
5434  Utot 372367
5435  Utot 323018
5436  Utot 309824
5437  Utot 357377
5438  Utot 400601
5439  Utot 401330
5440  Utot 285714
5441  Utot 303161
5442  Utot 313498
5443  Utot 309566
5444  Utot 336464
5445  Utot 314202
5446  Utot 266005
5447  Utot 299141
5448  Utot 293907
5449  Utot 348422
5450  Utot 340725
5451  Utot 385389
5452  Utot 375305
5453  Utot 378671
5454  Utot 309183
5455  Utot 317472
5456  Utot 374212
5457  Utot 340546
5458  Utot 338211
5459  Utot 362867
5460  Utot 358288
5461  Utot 348260
5462  Utot 295163
5463  Utot 368080
5464  Utot 375807
5465  Utot 348847
5466  Utot 301035
5467  Utot 296803
5468  Utot 347086
5469  Utot 397406
5470  Utot 326030
5471  Utot 365311
5472  Utot 303447
5473  Utot 312321
5474  Utot 363683
5475  Utot 288374
5476  Utot 358572
5477  Utot 323518
5478  Utot 332783
5479  Utot 412579
5480  Utot 352941
5481  Utot 318784
5482  Utot 337425
5483  Utot 327767
5484  Utot 350121
5485  Utot 330091
5486  Utot 334888
5487  Utot 306118
5488  Utot 368983
5489  Utot 334389
5490  Utot 299685
5491  Utot 345343
5492  Utot 304860
5493  Utot 327420
5494  Utot 328161
5495  Utot 320981
5496  Utot 310936
5497  Utot 400840
5498  Utot 314568
5499  Utot 322825
5500  Utot 312339
5501  Utot 333083
5502  Utot 307394
5503  Utot 379540
5504  Utot 327265
5505  Utot 383135
5506  Utot 341963
5507  Utot 387032
5508  Utot 306833
5509  Utot 353010
5510  Utot 335212
5511  Utot 354698
5512  Utot 287515
5513  Utot 313738
5514  Utot 328940
5515  Utot 309783
5516  Utot 309162
5517  Utot 275476
5518  Utot 328096
5519  Utot 306040
5520  Utot 295136
5521  Utot 396304
5522  Utot 291151
5523  Utot 332648
5524  Utot 404790
5525  Utot 298409
5526  Utot 351207
5527  Utot 341218
5528  Utot 333278
5529  Utot 390497
5530  Utot 353129
5531  Utot 300442
5532  Utot 315095
5533  Utot 404842
5534  Utot 322446
5535  Utot 326491
5536  Utot 316188
5537  Utot 291910
5538  Utot 336118
5539  Utot 329505
5540  Utot 334325
5541  Utot 331310
5542  Utot 381782
5543  Utot 324314
5544  Utot 366884
5545  Utot 336795
5546  Utot 375940
5547  Utot 311900
5548  Utot 324864
5549  Utot 374744
5550  Utot 302706
5551  Utot 298383
5552  Utot 285776
5553  Utot 326567
5554  Utot 294292
5555  Utot 297063
5556  Utot 323935
5557  Utot 323599
5558  Utot 334508
5559  Utot 410987
5560  Utot 369344
5561  Utot 313808
5562  Utot 321855
5563  Utot 288948
5564  Utot 376131
5565  Utot 329421
5566  Utot 304724
5567  Utot 374204
5568  Utot 365174
5569  Utot 349446
5570  Utot 364105
5571  Utot 341214
5572  Utot 328576
5573  Utot 321293
5574  Utot 418578
5575  Utot 317325
5576  Utot 387773
5577  Utot 316469
5578  Utot 343845
5579  Utot 394396
5580  Utot 278213
5581  Utot 306494
5582  Utot 344125
5583  Utot 285755
5584  Utot 347439
5585  Utot 346419
5586  Utot 321088
5587  Utot 298144
5588  Utot 305987
5589  Utot 362620
5590  Utot 350815
5591  Utot 329688
5592  Utot 313372
5593  Utot 349257
5594  Utot 353890
5595  Utot 351585
5596  Utot 356664
5597  Utot 317099
5598  Utot 300340
5599  Utot 339674
5600  Utot 352611
5601  Utot 282197
5602  Utot 409533
5603  Utot 317468
5604  Utot 307392
5605  Utot 368952
5606  Utot 361028
5607  Utot 327857
5608  Utot 286035
5609  Utot 295813
5610  Utot 336578
5611  Utot 291706
5612  Utot 330887
5613  Utot 378104
5614  Utot 317316
5615  Utot 342244
5616  Utot 322132
5617  Utot 457062
5618  Utot 345286
5619  Utot 299617
5620  Utot 361124
5621  Utot 312136
5622  Utot 321675
5623  Utot 302117
5624  Utot 354393
5625  Utot 396749
5626  Utot 318151
5627  Utot 283594
5628  Utot 373626
5629  Utot 328304
5630  Utot 314173
5631  Utot 298200
5632  Utot 359753
5633  Utot 332281
5634  Utot 404224
5635  Utot 308766
5636  Utot 322496
5637  Utot 418974
5638  Utot 345888
5639  Utot 301095
5640  Utot 315679
5641  Utot 367029
5642  Utot 350180
5643  Utot 304312
5644  Utot 375997
5645  Utot 295779
5646  Utot 295351
5647  Utot 299664
5648  Utot 336019
5649  Utot 387039
5650  Utot 279076
5651  Utot 381885
5652  Utot 327165
5653  Utot 348983
5654  Utot 319029
5655  Utot 297003
5656  Utot 297000
5657  Utot 285456
5658  Utot 342334
5659  Utot 297729
5660  Utot 337847
5661  Utot 358015
5662  Utot 341087
5663  Utot 293654
5664  Utot 282535
5665  Utot 347773
5666  Utot 366006
5667  Utot 291883
5668  Utot 312197
5669  Utot 390969
5670  Utot 289464
5671  Utot 351995
5672  Utot 319446
5673  Utot 308939
5674  Utot 340725
5675  Utot 364904
5676  Utot 415862
5677  Utot 277606
5678  Utot 304145
5679  Utot 345153
5680  Utot 282744
5681  Utot 353244
5682  Utot 301712
5683  Utot 332691
5684  Utot 277340
5685  Utot 357606
5686  Utot 361077
5687  Utot 335455
5688  Utot 346366
5689  Utot 348091
5690  Utot 299888
5691  Utot 327399
5692  Utot 307782
5693  Utot 379059
5694  Utot 308591
5695  Utot 263565
5696  Utot 326581
5697  Utot 336783
5698  Utot 308398
5699  Utot 364647
5700  Utot 301136
5701  Utot 311208
5702  Utot 309664
5703  Utot 351370
5704  Utot 306945
5705  Utot 338763
5706  Utot 274402
5707  Utot 327736
5708  Utot 428623
5709  Utot 309208
5710  Utot 296530
5711  Utot 325133
5712  Utot 374526
5713  Utot 299108
5714  Utot 346893
5715  Utot 321160
5716  Utot 321009
5717  Utot 339424
5718  Utot 295571
5719  Utot 315327
5720  Utot 315485
5721  Utot 314789
5722  Utot 298582
5723  Utot 322764
5724  Utot 331320
5725  Utot 336118
5726  Utot 313004
5727  Utot 300648
5728  Utot 328118
5729  Utot 337052
5730  Utot 339149
5731  Utot 307660
5732  Utot 313415
5733  Utot 372393
5734  Utot 321849
5735  Utot 329744
5736  Utot 324757
5737  Utot 344382
5738  Utot 356390
5739  Utot 311404
5740  Utot 367812
5741  Utot 304776
5742  Utot 330163
5743  Utot 283588
5744  Utot 326168
5745  Utot 330768
5746  Utot 278633
5747  Utot 319783
5748  Utot 388017
5749  Utot 315121
5750  Utot 303317
5751  Utot 322321
5752  Utot 304269
5753  Utot 303534
5754  Utot 410114
5755  Utot 311889
5756  Utot 292204
5757  Utot 308353
5758  Utot 343664
5759  Utot 303279
5760  Utot 336292
5761  Utot 335898
5762  Utot 345334
5763  Utot 373770
5764  Utot 352592
5765  Utot 331823
5766  Utot 390399
5767  Utot 308596
5768  Utot 307566
5769  Utot 305832
5770  Utot 305867
5771  Utot 311082
5772  Utot 335550
5773  Utot 326345
5774  Utot 285674
5775  Utot 382721
5776  Utot 345124
5777  Utot 328383
5778  Utot 300546
5779  Utot 409523
5780  Utot 302063
5781  Utot 431122
5782  Utot 314403
5783  Utot 300967
5784  Utot 323506
5785  Utot 308744
5786  Utot 327686
5787  Utot 280620
5788  Utot 359710
5789  Utot 329076
5790  Utot 293744
5791  Utot 302985
5792  Utot 318695
5793  Utot 317565
5794  Utot 316511
5795  Utot 282539
5796  Utot 394837
5797  Utot 307131
5798  Utot 299500
5799  Utot 324353
5800  Utot 368767
5801  Utot 376573
5802  Utot 364113
5803  Utot 340911
5804  Utot 363575
5805  Utot 268730
5806  Utot 295267
5807  Utot 268541
5808  Utot 332249
5809  Utot 328909
5810  Utot 315481
5811  Utot 355115
5812  Utot 376036
5813  Utot 315377
5814  Utot 382193
5815  Utot 443525
5816  Utot 318458
5817  Utot 364550
5818  Utot 329197
5819  Utot 285444
5820  Utot 367158
5821  Utot 289410
5822  Utot 388342
5823  Utot 375158
5824  Utot 298349
5825  Utot 376068
5826  Utot 437655
5827  Utot 357115
5828  Utot 333637
5829  Utot 322798
5830  Utot 340983
5831  Utot 349102
5832  Utot 383919
5833  Utot 301589
5834  Utot 304815
5835  Utot 275026
5836  Utot 328879
5837  Utot 465362
5838  Utot 340895
5839  Utot 317468
5840  Utot 292089
5841  Utot 305749
5842  Utot 333009
5843  Utot 390621
5844  Utot 459846
5845  Utot 314731
5846  Utot 317719
5847  Utot 365271
5848  Utot 309163
5849  Utot 333664
5850  Utot 307682
5851  Utot 427629
5852  Utot 323842
5853  Utot 312113
5854  Utot 295856
5855  Utot 418041
5856  Utot 309988
5857  Utot 348028
5858  Utot 297170
5859  Utot 328528
5860  Utot 306056
5861  Utot 302850
5862  Utot 303499
5863  Utot 306820
5864  Utot 330799
5865  Utot 367888
5866  Utot 291057
5867  Utot 314433
5868  Utot 273569
5869  Utot 333435
5870  Utot 336072
5871  Utot 340524
5872  Utot 298275
5873  Utot 326283
5874  Utot 317374
5875  Utot 337605
5876  Utot 282211
5877  Utot 307525
5878  Utot 313582
5879  Utot 330326
5880  Utot 299869
5881  Utot 351041
5882  Utot 320654
5883  Utot 313733
5884  Utot 323260
5885  Utot 319789
5886  Utot 317605
5887  Utot 277004
5888  Utot 293196
5889  Utot 298742
5890  Utot 330417
5891  Utot 338624
5892  Utot 351857
5893  Utot 335272
5894  Utot 328866
5895  Utot 304399
5896  Utot 369576
5897  Utot 318339
5898  Utot 333870
5899  Utot 356094
5900  Utot 287502
5901  Utot 306644
5902  Utot 340191
5903  Utot 392545
5904  Utot 317582
5905  Utot 340362
5906  Utot 313839
5907  Utot 287995
5908  Utot 290842
5909  Utot 352347
5910  Utot 324892
5911  Utot 321952
5912  Utot 370798
5913  Utot 418239
5914  Utot 279620
5915  Utot 313386
5916  Utot 302726
5917  Utot 340936
5918  Utot 290498
5919  Utot 337936
5920  Utot 311649
5921  Utot 327147
5922  Utot 371459
5923  Utot 311074
5924  Utot 353987
5925  Utot 310467
5926  Utot 303125
5927  Utot 361482
5928  Utot 342338
5929  Utot 380958
5930  Utot 334331
5931  Utot 347187
5932  Utot 363795
5933  Utot 382358
5934  Utot 370114
5935  Utot 405898
5936  Utot 335071
5937  Utot 305368
5938  Utot 381641
5939  Utot 368073
5940  Utot 329221
5941  Utot 384058
5942  Utot 354007
5943  Utot 351071
5944  Utot 279612
5945  Utot 357404
5946  Utot 333483
5947  Utot 286184
5948  Utot 311949
5949  Utot 338925
5950  Utot 374913
5951  Utot 286986
5952  Utot 301125
5953  Utot 272777
5954  Utot 287029
5955  Utot 311957
5956  Utot 326167
5957  Utot 393769
5958  Utot 292647
5959  Utot 288764
5960  Utot 278364
5961  Utot 343114
5962  Utot 367215
5963  Utot 322460
5964  Utot 359100
5965  Utot 300244
5966  Utot 308639
5967  Utot 316021
5968  Utot 322930
5969  Utot 365859
5970  Utot 355949
5971  Utot 347891
5972  Utot 334436
5973  Utot 320466
5974  Utot 331366
5975  Utot 358244
5976  Utot 351241
5977  Utot 323218
5978  Utot 333369
5979  Utot 345104
5980  Utot 382414
5981  Utot 321515
5982  Utot 334850
5983  Utot 378632
5984  Utot 342280
5985  Utot 321821
5986  Utot 339536
5987  Utot 325681
5988  Utot 331752
5989  Utot 342715
5990  Utot 323728
5991  Utot 361065
5992  Utot 326272
5993  Utot 313135
5994  Utot 274660
5995  Utot 309121
5996  Utot 329357
5997  Utot 314224
5998  Utot 345026
5999  Utot 380569
6000  Utot 334859
6001  Ntot 328407
6002  Ntot 331349
6003  Ntot 323358
6004  Ntot 310392
6005  Ntot 314010
6006  Ntot 310994
6007  Ntot 296059
6008  Ntot 293636
6009  Ntot 333368
6010  Ntot 360941
6011  Ntot 327260
6012  Ntot 295683
6013  Ntot 347724
6014  Ntot 347634
6015  Ntot 350867
6016  Ntot 358723
6017  Ntot 299358
6018  Ntot 334380
6019  Ntot 288729
6020  Ntot 346978
6021  Ntot 321194
6022  Ntot 286858
6023  Ntot 310209
6024  Ntot 292205
6025  Ntot 317943
6026  Ntot 333732
6027  Ntot 300986
6028  Ntot 321679
6029  Ntot 389103
6030  Ntot 344707
6031  Ntot 343839
6032  Ntot 324761
6033  Ntot 313557
6034  Ntot 333251
6035  Ntot 454165
6036  Ntot 350722
6037  Ntot 345390
6038  Ntot 336194
6039  Ntot 345264
6040  Ntot 362847
6041  Ntot 338913
6042  Ntot 341795
6043  Ntot 355602
6044  Ntot 299073
6045  Ntot 309670
6046  Ntot 409348
6047  Ntot 329796
6048  Ntot 345012
6049  Ntot 284567
6050  Ntot 284738
6051  Ntot 307265
6052  Ntot 416134
6053  Ntot 327967
6054  Ntot 331928
6055  Ntot 342112
6056  Ntot 366764
6057  Ntot 295186
6058  Ntot 314472
6059  Ntot 346440
6060  Ntot 284046
6061  Ntot 392750
6062  Ntot 298481
6063  Ntot 388883
6064  Ntot 334349
6065  Ntot 347644
6066  Ntot 330861
6067  Ntot 330805
6068  Ntot 332525
6069  Ntot 318246
6070  Ntot 347573
6071  Ntot 349106
6072  Ntot 329310
6073  Ntot 354218
6074  Ntot 317271
6075  Ntot 303183
6076  Ntot 342882
6077  Ntot 338125
6078  Ntot 311263
6079  Ntot 322521
6080  Ntot 291836
6081  Ntot 342376
6082  Ntot 315850
6083  Ntot 314319
6084  Ntot 306812
6085  Ntot 374313
6086  Ntot 371520
6087  Ntot 341323
6088  Ntot 327370
6089  Ntot 369748
6090  Ntot 311276
6091  Ntot 336246
6092  Ntot 277312
6093  Ntot 322052
6094  Ntot 327564
6095  Ntot 303667
6096  Ntot 319568
6097  Ntot 321821
6098  Ntot 314921
6099  Ntot 355931
6100  Ntot 286662
6101  Ntot 312405
6102  Ntot 349533
6103  Ntot 358786
6104  Ntot 332609
6105  Ntot 422554
6106  Ntot 305379
6107  Ntot 379052
6108  Ntot 292645
6109  Ntot 344993
6110  Ntot 331332
6111  Ntot 338596
6112  Ntot 325399
6113  Ntot 326290
6114  Ntot 343648
6115  Ntot 372203
6116  Ntot 331974
6117  Ntot 346818
6118  Ntot 301944
6119  Ntot 324036
6120  Ntot 348702
6121  Ntot 280667
6122  Ntot 286230
6123  Ntot 316685
6124  Ntot 406314
6125  Ntot 317744
6126  Ntot 330017
6127  Ntot 319967
6128  Ntot 344673
6129  Ntot 275993
6130  Ntot 348740
6131  Ntot 298563
6132  Ntot 317190
6133  Ntot 302557
6134  Ntot 327355
6135  Ntot 318677
6136  Ntot 386001
6137  Ntot 312264
6138  Ntot 305981
6139  Ntot 333545
6140  Ntot 353243
6141  Ntot 356042
6142  Ntot 350870
6143  Ntot 321791
6144  Ntot 336062
6145  Ntot 382098
6146  Ntot 353267
6147  Ntot 323736
6148  Ntot 372293
6149  Ntot 344669
6150  Ntot 366095
6151  Ntot 321987
6152  Ntot 332459
6153  Ntot 343811
6154  Ntot 361186
6155  Ntot 306889
6156  Ntot 320846
6157  Ntot 359877
6158  Ntot 295436
6159  Ntot 351287
6160  Ntot 342362
6161  Ntot 336683
6162  Ntot 310006
6163  Ntot 406873
6164  Ntot 300604
6165  Ntot 328737
6166  Ntot 362637
6167  Ntot 303473
6168  Ntot 312058
6169  Ntot 334709
6170  Ntot 360506
6171  Ntot 325032
6172  Ntot 318985
6173  Ntot 309286
6174  Ntot 380886
6175  Ntot 317455
6176  Ntot 308762
6177  Ntot 355788
6178  Ntot 340096
6179  Ntot 340905
6180  Ntot 399412
6181  Ntot 348754
6182  Ntot 317875
6183  Ntot 366952
6184  Ntot 360047
6185  Ntot 369207
6186  Ntot 314071
6187  Ntot 328038
6188  Ntot 308840
6189  Ntot 371757
6190  Ntot 304728
6191  Ntot 330035
6192  Ntot 329293
6193  Ntot 339386
6194  Ntot 331336
6195  Ntot 340079
6196  Ntot 386060
6197  Ntot 295014
6198  Ntot 405543
6199  Ntot 349920
6200  Ntot 350780
6201  Ntot 314514
6202  Ntot 346668
6203  Ntot 308764
6204  Ntot 323189
6205  Ntot 297452
6206  Ntot 283004
6207  Ntot 341645
6208  Ntot 315907
6209  Ntot 358967
6210  Ntot 314361
6211  Ntot 357940
6212  Ntot 281944
6213  Ntot 355882
6214  Ntot 332494
6215  Ntot 405268
6216  Ntot 328020
6217  Ntot 313649
6218  Ntot 328651
6219  Ntot 331169
6220  Ntot 304794
6221  Ntot 348054
6222  Ntot 288809
6223  Ntot 317163
6224  Ntot 337541
6225  Ntot 321204
6226  Ntot 345706
6227  Ntot 369626
6228  Ntot 337222
6229  Ntot 301726
6230  Ntot 290469
6231  Ntot 326463
6232  Ntot 300813
6233  Ntot 313809
6234  Ntot 337858
6235  Ntot 315452
6236  Ntot 307798
6237  Ntot 411653
6238  Ntot 351141
6239  Ntot 333567
6240  Ntot 369587
6241  Ntot 388112
6242  Ntot 354821
6243  Ntot 272867
6244  Ntot 395798
6245  Ntot 310750
6246  Ntot 397235
6247  Ntot 409278
6248  Ntot 361052
6249  Ntot 347343
6250  Ntot 315124
6251  Ntot 335709
6252  Ntot 441112
6253  Ntot 436015
6254  Ntot 324373
6255  Ntot 341935
6256  Ntot 331023
6257  Ntot 328465
6258  Ntot 281104
6259  Ntot 329714
6260  Ntot 316163
6261  Ntot 364161
6262  Ntot 278791
6263  Ntot 301622
6264  Ntot 328149
6265  Ntot 361840
6266  Ntot 310321
6267  Ntot 310462
6268  Ntot 299292
6269  Ntot 322318
6270  Ntot 306972
6271  Ntot 310353
6272  Ntot 378921
6273  Ntot 312001
6274  Ntot 354087
6275  Ntot 318173
6276  Ntot 336247
6277  Ntot 316549
6278  Ntot 345463
6279  Ntot 345738
6280  Ntot 334670
6281  Ntot 324695
6282  Ntot 301004
6283  Ntot 376861
6284  Ntot 370129
6285  Ntot 412114
6286  Ntot 318948
6287  Ntot 301857
6288  Ntot 338256
6289  Ntot 339289
6290  Ntot 301272
6291  Ntot 334320
6292  Ntot 475008
6293  Ntot 354471
6294  Ntot 337264
6295  Ntot 334394
6296  Ntot 363701
6297  Ntot 322757
6298  Ntot 309718
6299  Ntot 337645
6300  Ntot 306672
6301  Ntot 323297
6302  Ntot 424636
6303  Ntot 345390
6304  Ntot 326516
6305  Ntot 342967
6306  Ntot 358944
6307  Ntot 410672
6308  Ntot 347322
6309  Ntot 305919
6310  Ntot 287575
6311  Ntot 362781
6312  Ntot 346308
6313  Ntot 323533
6314  Ntot 324816
6315  Ntot 289817
6316  Ntot 338029
6317  Ntot 345159
6318  Ntot 344640
6319  Ntot 325515
6320  Ntot 336295
6321  Ntot 398108
6322  Ntot 353433
6323  Ntot 337039
6324  Ntot 335895
6325  Ntot 380987
6326  Ntot 321535
6327  Ntot 369135
6328  Ntot 314720
6329  Ntot 306506
6330  Ntot 385923
6331  Ntot 316589
6332  Ntot 319535
6333  Ntot 312790
6334  Ntot 360353
6335  Ntot 354775
6336  Ntot 352995
6337  Ntot 343952
6338  Ntot 295058
6339  Ntot 299793
6340  Ntot 320992
6341  Ntot 363126
6342  Ntot 312522
6343  Ntot 345779
6344  Ntot 358179
6345  Ntot 383395
6346  Ntot 307447
6347  Ntot 356761
6348  Ntot 322142
6349  Ntot 358218
6350  Ntot 302907
6351  Ntot 360369
6352  Ntot 325119
6353  Ntot 323740
6354  Ntot 384752
6355  Ntot 301764
6356  Ntot 313851
6357  Ntot 311991
6358  Ntot 351544
6359  Ntot 318519
6360  Ntot 354395
6361  Ntot 344317
6362  Ntot 317890
6363  Ntot 317949
6364  Ntot 387443
6365  Ntot 324419
6366  Ntot 302479
6367  Ntot 341583
6368  Ntot 281452
6369  Ntot 326248
6370  Ntot 403742
6371  Ntot 318987
6372  Ntot 360696
6373  Ntot 386248
6374  Ntot 374102
6375  Ntot 337920
6376  Ntot 328968
6377  Ntot 381330
6378  Ntot 378741
6379  Ntot 353456
6380  Ntot 333685
6381  Ntot 306822
6382  Ntot 352999
6383  Ntot 278897
6384  Ntot 312831
6385  Ntot 354073
6386  Ntot 341037
6387  Ntot 325700
6388  Ntot 315840
6389  Ntot 339847
6390  Ntot 307439
6391  Ntot 298800
6392  Ntot 351588
6393  Ntot 325882
6394  Ntot 348345
6395  Ntot 307011
6396  Ntot 293290
6397  Ntot 302734
6398  Ntot 362345
6399  Ntot 324770
6400  Ntot 307702
6401  Ntot 369412
6402  Ntot 305657
6403  Ntot 295340
6404  Ntot 331581
6405  Ntot 423243
6406  Ntot 408345
6407  Ntot 332461
6408  Ntot 313862
6409  Ntot 321433
6410  Ntot 314838
6411  Ntot 328503
6412  Ntot 301981
6413  Ntot 315267
6414  Ntot 321892
6415  Ntot 302891
6416  Ntot 370949
6417  Ntot 342328
6418  Ntot 294668
6419  Ntot 340808
6420  Ntot 424807
6421  Ntot 327099
6422  Ntot 354493
6423  Ntot 284943
6424  Ntot 282610
6425  Ntot 313986
6426  Ntot 360589
6427  Ntot 306467
6428  Ntot 335872
6429  Ntot 318492
6430  Ntot 440002
6431  Ntot 292471
6432  Ntot 364171
6433  Ntot 333105
6434  Ntot 352329
6435  Ntot 319311
6436  Ntot 306940
6437  Ntot 365581
6438  Ntot 371120
6439  Ntot 343664
6440  Ntot 374035
6441  Ntot 345824
6442  Ntot 340306
6443  Ntot 338947
6444  Ntot 326686
6445  Ntot 303588
6446  Ntot 318048
6447  Ntot 314626
6448  Ntot 389866
6449  Ntot 287180
6450  Ntot 382507
6451  Ntot 338416
6452  Ntot 316326
6453  Ntot 331912
6454  Ntot 333925
6455  Ntot 340881
6456  Ntot 309956
6457  Ntot 302041
6458  Ntot 330218
6459  Ntot 300486
6460  Ntot 393817
6461  Ntot 338150
6462  Ntot 278365
6463  Ntot 288812
6464  Ntot 338108
6465  Ntot 373118
6466  Ntot 326893
6467  Ntot 342187
6468  Ntot 391708
6469  Ntot 346414
6470  Ntot 306763
6471  Ntot 335913
6472  Ntot 327560
6473  Ntot 351271
6474  Ntot 394824
6475  Ntot 314371
6476  Ntot 315256
6477  Ntot 296461
6478  Ntot 401349
6479  Ntot 355376
6480  Ntot 329026
6481  Ntot 278723
6482  Ntot 356495
6483  Ntot 305729
6484  Ntot 332842
6485  Ntot 287297
6486  Ntot 321632
6487  Ntot 415618
6488  Ntot 336235
6489  Ntot 333838
6490  Ntot 418071
6491  Ntot 365357
6492  Ntot 365412
6493  Ntot 349207
6494  Ntot 324865
6495  Ntot 380764
6496  Ntot 293896
6497  Ntot 346750
6498  Ntot 357272
6499  Ntot 361097
6500  Ntot 315655
6501  Ntot 358567
6502  Ntot 317276
6503  Ntot 412233
6504  Ntot 301889
6505  Ntot 314041
6506  Ntot 323216
6507  Ntot 327532
6508  Ntot 335633
6509  Ntot 363410
6510  Ntot 357333
6511  Ntot 322872
6512  Ntot 336564
6513  Ntot 310611
6514  Ntot 351917
6515  Ntot 326396
6516  Ntot 348763
6517  Ntot 282729
6518  Ntot 289851
6519  Ntot 339777
6520  Ntot 324987
6521  Ntot 374799
6522  Ntot 341762
6523  Ntot 360985
6524  Ntot 372086
6525  Ntot 345596
6526  Ntot 306572
6527  Ntot 298326
6528  Ntot 306756
6529  Ntot 314601
6530  Ntot 349944
6531  Ntot 294367
6532  Ntot 417976
6533  Ntot 391566
6534  Ntot 337227
6535  Ntot 317227
6536  Ntot 323131
6537  Ntot 292016
6538  Ntot 360567
6539  Ntot 338010
6540  Ntot 328290
6541  Ntot 351786
6542  Ntot 336751
6543  Ntot 288914
6544  Ntot 317918
6545  Ntot 350842
6546  Ntot 335075
6547  Ntot 283571
6548  Ntot 314168
6549  Ntot 333325
6550  Ntot 483894
6551  Ntot 362207
6552  Ntot 309525
6553  Ntot 320063
6554  Ntot 333175
6555  Ntot 344629
6556  Ntot 367243
6557  Ntot 302019
6558  Ntot 314293
6559  Ntot 337710
6560  Ntot 339596
6561  Ntot 378060
6562  Ntot 357823
6563  Ntot 353789
6564  Ntot 301928
6565  Ntot 329274
6566  Ntot 305795
6567  Ntot 381788
6568  Ntot 364652
6569  Ntot 306919
6570  Ntot 274479
6571  Ntot 313484
6572  Ntot 329923
6573  Ntot 362518
6574  Ntot 364349
6575  Ntot 352125
6576  Ntot 296529
6577  Ntot 339412
6578  Ntot 304420
6579  Ntot 356300
6580  Ntot 326545
6581  Ntot 285728
6582  Ntot 353481
6583  Ntot 342476
6584  Ntot 300093
6585  Ntot 345586
6586  Ntot 351879
6587  Ntot 379760
6588  Ntot 295405
6589  Ntot 321837
6590  Ntot 304384
6591  Ntot 296053
6592  Ntot 318386
6593  Ntot 297853
6594  Ntot 314365
6595  Ntot 322735
6596  Ntot 354145
6597  Ntot 322931
6598  Ntot 387414
6599  Ntot 336084
6600  Ntot 364562
6601  Ntot 323013
6602  Ntot 314922
6603  Ntot 358717
6604  Ntot 374445
6605  Ntot 299680
6606  Ntot 349961
6607  Ntot 307441
6608  Ntot 330051
6609  Ntot 312889
6610  Ntot 337800
6611  Ntot 374952
6612  Ntot 357327
6613  Ntot 356252
6614  Ntot 307616
6615  Ntot 309410
6616  Ntot 373247
6617  Ntot 326987
6618  Ntot 292906
6619  Ntot 336267
6620  Ntot 321315
6621  Ntot 359250
6622  Ntot 379655
6623  Ntot 295709
6624  Ntot 363322
6625  Ntot 307871
6626  Ntot 347991
6627  Ntot 365630
6628  Ntot 336819
6629  Ntot 355221
6630  Ntot 298885
6631  Ntot 356770
6632  Ntot 323617
6633  Ntot 356843
6634  Ntot 353530
6635  Ntot 311944
6636  Ntot 358256
6637  Ntot 334953
6638  Ntot 338125
6639  Ntot 322069
6640  Ntot 308328
6641  Ntot 322485
6642  Ntot 331335
6643  Ntot 311297
6644  Ntot 311733
6645  Ntot 311025
6646  Ntot 308968
6647  Ntot 325917
6648  Ntot 402486
6649  Ntot 355993
6650  Ntot 312100
6651  Ntot 302429
6652  Ntot 319354
6653  Ntot 398392
6654  Ntot 324008
6655  Ntot 379577
6656  Ntot 309812
6657  Ntot 341065
6658  Ntot 364009
6659  Ntot 445031
6660  Ntot 346364
6661  Ntot 413222
6662  Ntot 303956
6663  Ntot 289678
6664  Ntot 322039
6665  Ntot 301844
6666  Ntot 323006
6667  Ntot 332476
6668  Ntot 352371
6669  Ntot 308066
6670  Ntot 352493
6671  Ntot 355881
6672  Ntot 351734
6673  Ntot 293822
6674  Ntot 354721
6675  Ntot 273832
6676  Ntot 403283
6677  Ntot 321753
6678  Ntot 361799
6679  Ntot 344594
6680  Ntot 311217
6681  Ntot 354066
6682  Ntot 319142
6683  Ntot 322281
6684  Ntot 352405
6685  Ntot 387738
6686  Ntot 306035
6687  Ntot 335216
6688  Ntot 301117
6689  Ntot 288009
6690  Ntot 329270
6691  Ntot 343050
6692  Ntot 324777
6693  Ntot 350099
6694  Ntot 291152
6695  Ntot 303823
6696  Ntot 293589
6697  Ntot 338266
6698  Ntot 294422
6699  Ntot 309722
6700  Ntot 329859
6701  Ntot 400925
6702  Ntot 318625
6703  Ntot 316101
6704  Ntot 367412
6705  Ntot 301331
6706  Ntot 306600
6707  Ntot 392438
6708  Ntot 308226
6709  Ntot 390761
6710  Ntot 373905
6711  Ntot 324838
6712  Ntot 294466
6713  Ntot 321570
6714  Ntot 287649
6715  Ntot 317515
6716  Ntot 318372
6717  Ntot 373901
6718  Ntot 382151
6719  Ntot 320319
6720  Ntot 272815
6721  Ntot 364135
6722  Ntot 397306
6723  Ntot 334117
6724  Ntot 310047
6725  Ntot 323735
6726  Ntot 312628
6727  Ntot 330036
6728  Ntot 386774
6729  Ntot 308058
6730  Ntot 334586
6731  Ntot 281806
6732  Ntot 350209
6733  Ntot 442127
6734  Ntot 419455
6735  Ntot 320245
6736  Ntot 379538
6737  Ntot 318536
6738  Ntot 326051
6739  Ntot 333302
6740  Ntot 335371
6741  Ntot 352428
6742  Ntot 396869
6743  Ntot 326425
6744  Ntot 321666
6745  Ntot 346956
6746  Ntot 303427
6747  Ntot 304615
6748  Ntot 368397
6749  Ntot 353878
6750  Ntot 340143
6751  Ntot 281124
6752  Ntot 308740
6753  Ntot 347567
6754  Ntot 302555
6755  Ntot 332479
6756  Ntot 322137
6757  Ntot 362484
6758  Ntot 395294
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6760  Ntot 380160
6761  Ntot 358456
6762  Ntot 341179
6763  Ntot 336446
6764  Ntot 390197
6765  Ntot 343335
6766  Ntot 348878
6767  Ntot 387308
6768  Ntot 313694
6769  Ntot 394810
6770  Ntot 338434
6771  Ntot 405759
6772  Ntot 336829
6773  Ntot 290203
6774  Ntot 312804
6775  Ntot 351836
6776  Ntot 337070
6777  Ntot 409989
6778  Ntot 310761
6779  Ntot 343077
6780  Ntot 345685
6781  Ntot 396193
6782  Ntot 308310
6783  Ntot 414435
6784  Ntot 339152
6785  Ntot 320210
6786  Ntot 310894
6787  Ntot 384796
6788  Ntot 301224
6789  Ntot 363696
6790  Ntot 360662
6791  Ntot 332884
6792  Ntot 344187
6793  Ntot 345682
6794  Ntot 324887
6795  Ntot 288455
6796  Ntot 310259
6797  Ntot 286945
6798  Ntot 299277
6799  Ntot 344845
6800  Ntot 328093
6801  Ntot 332926
6802  Ntot 337062
6803  Ntot 381140
6804  Ntot 310766
6805  Ntot 345205
6806  Ntot 319631
6807  Ntot 304703
6808  Ntot 309057
6809  Ntot 316762
6810  Ntot 325471
6811  Ntot 277534
6812  Ntot 297918
6813  Ntot 291843
6814  Ntot 316727
6815  Ntot 315138
6816  Ntot 364333
6817  Ntot 342950
6818  Ntot 338161
6819  Ntot 352761
6820  Ntot 395504
6821  Ntot 326350
6822  Ntot 408913
6823  Ntot 348401
6824  Ntot 324057
6825  Ntot 301675
6826  Ntot 317969
6827  Ntot 335111
6828  Ntot 353469
6829  Ntot 386507
6830  Ntot 355643
6831  Ntot 282056
6832  Ntot 310130
6833  Ntot 366925
6834  Ntot 323820
6835  Ntot 339149
6836  Ntot 359553
6837  Ntot 318572
6838  Ntot 315251
6839  Ntot 323380
6840  Ntot 341213
6841  Ntot 412521
6842  Ntot 313441
6843  Ntot 359125
6844  Ntot 371198
6845  Ntot 302188
6846  Ntot 411874
6847  Ntot 347130
6848  Ntot 297185
6849  Ntot 361422
6850  Ntot 320287
6851  Ntot 322436
6852  Ntot 312927
6853  Ntot 342486
6854  Ntot 295541
6855  Ntot 366668
6856  Ntot 321113
6857  Ntot 347608
6858  Ntot 338426
6859  Ntot 331124
6860  Ntot 304500
6861  Ntot 360518
6862  Ntot 380865
6863  Ntot 285882
6864  Ntot 419310
6865  Ntot 288726
6866  Ntot 315398
6867  Ntot 444310
6868  Ntot 303310
6869  Ntot 323839
6870  Ntot 318395
6871  Ntot 429381
6872  Ntot 335187
6873  Ntot 385789
6874  Ntot 313233
6875  Ntot 348194
6876  Ntot 326377
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6878  Ntot 303259
6879  Ntot 359211
6880  Ntot 346498
6881  Ntot 408238
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6883  Ntot 377255
6884  Ntot 327141
6885  Ntot 314542
6886  Ntot 310400
6887  Ntot 353879
6888  Ntot 349382
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6890  Ntot 337423
6891  Ntot 330052
6892  Ntot 292554
6893  Ntot 325463
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6895  Ntot 351102
6896  Ntot 352104
6897  Ntot 446514
6898  Ntot 344811
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6900  Ntot 324617
6901  Ntot 322072
6902  Ntot 328445
6903  Ntot 342693
6904  Ntot 395284
6905  Ntot 294635
6906  Ntot 376920
6907  Ntot 345507
6908  Ntot 323526
6909  Ntot 347623
6910  Ntot 315745
6911  Ntot 351154
6912  Ntot 351335
6913  Ntot 358725
6914  Ntot 327443
6915  Ntot 313634
6916  Ntot 367740
6917  Ntot 427050
6918  Ntot 369792
6919  Ntot 428994
6920  Ntot 344106
6921  Ntot 318465
6922  Ntot 309320
6923  Ntot 363760
6924  Ntot 322671
6925  Ntot 333342
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6927  Ntot 357200
6928  Ntot 307875
6929  Ntot 340664
6930  Ntot 354883
6931  Ntot 350290
6932  Ntot 371892
6933  Ntot 335349
6934  Ntot 403990
6935  Ntot 328917
6936  Ntot 380625
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6938  Ntot 328200
6939  Ntot 369046
6940  Ntot 340805
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6942  Ntot 297924
6943  Ntot 322995
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6945  Ntot 319569
6946  Ntot 349795
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6948  Ntot 338748
6949  Ntot 309687
6950  Ntot 363612
6951  Ntot 349508
6952  Ntot 288051
6953  Ntot 303231
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6955  Ntot 361531
6956  Ntot 319211
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6958  Ntot 323536
6959  Ntot 379243
6960  Ntot 378900
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6962  Ntot 322626
6963  Ntot 297105
6964  Ntot 316134
6965  Ntot 318313
6966  Ntot 320965
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6968  Ntot 332207
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6970  Ntot 293142
6971  Ntot 332631
6972  Ntot 321963
6973  Ntot 298198
6974  Ntot 313674
6975  Ntot 363145
6976  Ntot 315997
6977  Ntot 330103
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6979  Ntot 335912
6980  Ntot 330594
6981  Ntot 328083
6982  Ntot 336059
6983  Ntot 455529
6984  Ntot 370990
6985  Ntot 360807
6986  Ntot 300755
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6988  Ntot 336520
6989  Ntot 361260
6990  Ntot 386876
6991  Ntot 318098
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6993  Ntot 353796
6994  Ntot 339624
6995  Ntot 322587
6996  Ntot 330291
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6998  Ntot 293697
6999  Ntot 360771
7000  Ntot 308698
7001  Ntot 349723
7002  Ntot 294112
7003  Ntot 316221
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7005  Ntot 351545
7006  Ntot 375347
7007  Ntot 297535
7008  Ntot 401184
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7010  Ntot 347198
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7012  Ntot 310162
7013  Ntot 329101
7014  Ntot 334407
7015  Ntot 309464
7016  Ntot 329797
7017  Ntot 304073
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7019  Ntot 335646
7020  Ntot 324634
7021  Ntot 318003
7022  Ntot 383562
7023  Ntot 324893
7024  Ntot 356767
7025  Ntot 365117
7026  Ntot 302172
7027  Ntot 382374
7028  Ntot 287080
7029  Ntot 357746
7030  Ntot 299293
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7032  Ntot 372454
7033  Ntot 336485
7034  Ntot 279957
7035  Ntot 370414
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7038  Ntot 350669
7039  Ntot 348766
7040  Ntot 391518
7041  Ntot 404669
7042  Ntot 321642
7043  Ntot 324339
7044  Ntot 293069
7045  Ntot 370843
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7047  Ntot 288862
7048  Ntot 313551
7049  Ntot 356891
7050  Ntot 389399
7051  Ntot 290295
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7053  Ntot 323453
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7055  Ntot 352543
7056  Ntot 339477
7057  Ntot 311048
7058  Ntot 286771
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7060  Ntot 299156
7061  Ntot 316318
7062  Ntot 335430
7063  Ntot 359797
7064  Ntot 359803
7065  Ntot 362688
7066  Ntot 311751
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7068  Ntot 336225
7069  Ntot 383534
7070  Ntot 330510
7071  Ntot 334268
7072  Ntot 390202
7073  Ntot 394234
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7075  Ntot 309844
7076  Ntot 337898
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7078  Ntot 307060
7079  Ntot 336208
7080  Ntot 364550
7081  Ntot 414083
7082  Ntot 380137
7083  Ntot 319337
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7085  Ntot 311241
7086  Ntot 306290
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7088  Ntot 358810
7089  Ntot 340809
7090  Ntot 318757
7091  Ntot 315681
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7093  Ntot 374604
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7095  Ntot 415775
7096  Ntot 362168
7097  Ntot 344738
7098  Ntot 379675
7099  Ntot 292860
7100  Ntot 319269
7101  Ntot 292387
7102  Ntot 345848
7103  Ntot 310461
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7106  Ntot 316711
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7108  Ntot 298009
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7110  Ntot 342249
7111  Ntot 370030
7112  Ntot 335407
7113  Ntot 342327
7114  Ntot 277527
7115  Ntot 323307
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7118  Ntot 434330
7119  Ntot 362576
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7126  Ntot 337821
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7128  Ntot 363602
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7136  Ntot 377385
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7210  Ntot 294362
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7249  Ntot 329716
7250  Ntot 328013
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7252  Ntot 366334
7253  Ntot 311377
7254  Ntot 352984
7255  Ntot 288638
7256  Ntot 383439
7257  Ntot 333677
7258  Ntot 295336
7259  Ntot 327118
7260  Ntot 363907
7261  Ntot 344437
7262  Ntot 377686
7263  Ntot 336838
7264  Ntot 307573
7265  Ntot 289779
7266  Ntot 299908
7267  Ntot 286240
7268  Ntot 303917
7269  Ntot 371829
7270  Ntot 274384
7271  Ntot 396829
7272  Ntot 308778
7273  Ntot 309203
7274  Ntot 305832
7275  Ntot 358583
7276  Ntot 373117
7277  Ntot 294897
7278  Ntot 377864
7279  Ntot 298493
7280  Ntot 333069
7281  Ntot 325393
7282  Ntot 321341
7283  Ntot 291815
7284  Ntot 310208
7285  Ntot 395005
7286  Ntot 351316
7287  Ntot 318092
7288  Ntot 305460
7289  Ntot 318114
7290  Ntot 335657
7291  Ntot 326066
7292  Ntot 297767
7293  Ntot 359187
7294  Ntot 348318
7295  Ntot 333991
7296  Ntot 381018
7297  Ntot 325123
7298  Ntot 331973
7299  Ntot 374251
7300  Ntot 334893
7301  Ntot 414899
7302  Ntot 322548
7303  Ntot 310694
7304  Ntot 359074
7305  Ntot 378157
7306  Ntot 315935
7307  Ntot 347165
7308  Ntot 300749
7309  Ntot 345716
7310  Ntot 347729
7311  Ntot 298019
7312  Ntot 300786
7313  Ntot 348277
7314  Ntot 331420
7315  Ntot 372876
7316  Ntot 340821
7317  Ntot 335147
7318  Ntot 336291
7319  Ntot 307095
7320  Ntot 395241
7321  Ntot 311590
7322  Ntot 324560
7323  Ntot 316848
7324  Ntot 390860
7325  Ntot 336377
7326  Ntot 332625
7327  Ntot 315259
7328  Ntot 373589
7329  Ntot 319216
7330  Ntot 323320
7331  Ntot 398048
7332  Ntot 356478
7333  Ntot 335719
7334  Ntot 320995
7335  Ntot 347745
7336  Ntot 345984
7337  Ntot 338841
7338  Ntot 319100
7339  Ntot 311010
7340  Ntot 356424
7341  Ntot 398301
7342  Ntot 301949
7343  Ntot 313854
7344  Ntot 280480
7345  Ntot 298154
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7347  Ntot 344817
7348  Ntot 381180
7349  Ntot 373094
7350  Ntot 349299
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7352  Ntot 327096
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7354  Ntot 349567
7355  Ntot 301869
7356  Ntot 402006
7357  Ntot 406921
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7360  Ntot 354344
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7362  Ntot 321957
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7364  Ntot 311285
7365  Ntot 330741
7366  Ntot 316336
7367  Ntot 293076
7368  Ntot 298931
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7370  Ntot 332259
7371  Ntot 313895
7372  Ntot 337977
7373  Ntot 271983
7374  Ntot 387488
7375  Ntot 379825
7376  Ntot 308320
7377  Ntot 367773
7378  Ntot 383827
7379  Ntot 317309
7380  Ntot 380074
7381  Ntot 352850
7382  Ntot 307117
7383  Ntot 325673
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7386  Ntot 368029
7387  Ntot 386348
7388  Ntot 324474
7389  Ntot 387375
7390  Ntot 397843
7391  Ntot 412602
7392  Ntot 329554
7393  Ntot 328363
7394  Ntot 324936
7395  Ntot 388344
7396  Ntot 307902
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7398  Ntot 321813
7399  Ntot 357145
7400  Ntot 303549
7401  Ntot 316948
7402  Ntot 455511
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7404  Ntot 360643
7405  Ntot 360823
7406  Ntot 307932
7407  Ntot 302633
7408  Ntot 328435
7409  Ntot 306010
7410  Ntot 323136
7411  Ntot 301443
7412  Ntot 347066
7413  Ntot 337380
7414  Ntot 362143
7415  Ntot 348331
7416  Ntot 343953
7417  Ntot 315697
7418  Ntot 287988
7419  Ntot 352408
7420  Ntot 322015
7421  Ntot 320315
7422  Ntot 392247
7423  Ntot 338480
7424  Ntot 329989
7425  Ntot 383140
7426  Ntot 305482
7427  Ntot 341723
7428  Ntot 302263
7429  Ntot 303529
7430  Ntot 316770
7431  Ntot 298212
7432  Ntot 322121
7433  Ntot 321747
7434  Ntot 413207
7435  Ntot 297516
7436  Ntot 343257
7437  Ntot 343678
7438  Ntot 416743
7439  Ntot 318871
7440  Ntot 282927
7441  Ntot 328661
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7443  Ntot 343606
7444  Ntot 351911
7445  Ntot 297482
7446  Ntot 333053
7447  Ntot 354887
7448  Ntot 330043
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7450  Ntot 356690
7451  Ntot 331275
7452  Ntot 377951
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7455  Ntot 356306
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7457  Ntot 303515
7458  Ntot 358510
7459  Ntot 290092
7460  Ntot 309412
7461  Ntot 337912
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7463  Ntot 319982
7464  Ntot 328586
7465  Ntot 305320
7466  Ntot 337084
7467  Ntot 355895
7468  Ntot 314864
7469  Ntot 381444
7470  Ntot 425350
7471  Ntot 339318
7472  Ntot 305753
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7474  Ntot 361388
7475  Ntot 319366
7476  Ntot 367287
7477  Ntot 310899
7478  Ntot 318581
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7480  Ntot 346124
7481  Ntot 307191
7482  Ntot 297023
7483  Ntot 413474
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7485  Ntot 333243
7486  Ntot 324430
7487  Ntot 328123
7488  Ntot 330949
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7490  Ntot 313037
7491  Ntot 415998
7492  Ntot 326867
7493  Ntot 294023
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7495  Ntot 445887
7496  Ntot 306503
7497  Ntot 423591
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7499  Ntot 354281
7500  Ntot 359127
7501  Ntot 305334
7502  Ntot 408107
7503  Ntot 347912
7504  Ntot 333133
7505  Ntot 302761
7506  Ntot 339227
7507  Ntot 346429
7508  Ntot 344541
7509  Ntot 341472
7510  Ntot 306946
7511  Ntot 329317
7512  Ntot 382926
7513  Ntot 412801
7514  Ntot 353792
7515  Ntot 328362
7516  Ntot 305436
7517  Ntot 272404
7518  Ntot 357715
7519  Ntot 339422
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7521  Ntot 307518
7522  Ntot 325938
7523  Ntot 294277
7524  Ntot 362729
7525  Ntot 313475
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7527  Ntot 378592
7528  Ntot 315780
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7530  Ntot 337704
7531  Ntot 372151
7532  Ntot 380346
7533  Ntot 298122
7534  Ntot 307928
7535  Ntot 376783
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7537  Ntot 310824
7538  Ntot 372925
7539  Ntot 301184
7540  Ntot 328531
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7542  Ntot 288797
7543  Ntot 360964
7544  Ntot 346986
7545  Ntot 335443
7546  Ntot 298096
7547  Ntot 345895
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7549  Ntot 397032
7550  Ntot 377988
7551  Ntot 370386
7552  Ntot 284571
7553  Ntot 320480
7554  Ntot 319956
7555  Ntot 338592
7556  Ntot 331774
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7558  Ntot 325885
7559  Ntot 420903
7560  Ntot 320204
7561  Ntot 336398
7562  Ntot 302912
7563  Ntot 336009
7564  Ntot 329434
7565  Ntot 332931
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7567  Ntot 362949
7568  Ntot 377278
7569  Ntot 435952
7570  Ntot 329299
7571  Ntot 325734
7572  Ntot 311413
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7574  Ntot 336940
7575  Ntot 329427
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7577  Ntot 299015
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7579  Ntot 288004
7580  Ntot 270146
7581  Ntot 372037
7582  Ntot 370735
7583  Ntot 368592
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7585  Ntot 304579
7586  Ntot 335148
7587  Ntot 304852
7588  Ntot 325619
7589  Ntot 315356
7590  Ntot 404485
7591  Ntot 359747
7592  Ntot 296473
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7594  Ntot 322244
7595  Ntot 333741
7596  Ntot 334627
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7598  Ntot 302599
7599  Ntot 333205
7600  Ntot 384376
7601  Ntot 303866
7602  Ntot 340118
7603  Ntot 329242
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7606  Ntot 312576
7607  Ntot 339851
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7609  Ntot 411972
7610  Ntot 343445
7611  Ntot 351091
7612  Ntot 394926
7613  Ntot 336200
7614  Ntot 337203
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7616  Ntot 341681
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7618  Ntot 328830
7619  Ntot 363048
7620  Ntot 304038
7621  Ntot 319196
7622  Ntot 325291
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7624  Ntot 390352
7625  Ntot 324412
7626  Ntot 416576
7627  Ntot 317384
7628  Ntot 329186
7629  Ntot 338355
7630  Ntot 349603
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7632  Ntot 330407
7633  Ntot 296935
7634  Ntot 343228
7635  Ntot 352907
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7637  Ntot 333905
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7639  Ntot 273512
7640  Ntot 314860
7641  Ntot 360316
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7649  Ntot 330877
7650  Ntot 334119
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7654  Ntot 316933
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7656  Ntot 329726
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7660  Ntot 378409
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7664  Ntot 325680
7665  Ntot 304642
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7667  Ntot 381299
7668  Ntot 360518
7669  Ntot 334860
7670  Ntot 318863
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7675  Ntot 382225
7676  Ntot 355252
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7680  Ntot 353654
7681  Ntot 332612
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7683  Ntot 425492
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7685  Ntot 328746
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7688  Ntot 300081
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7699  Ntot 346313
7700  Ntot 282601
7701  Ntot 304912
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7705  Ntot 426295
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7710  Ntot 331173
7711  Ntot 340518
7712  Ntot 338676
7713  Ntot 300008
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7715  Ntot 339859
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7719  Ntot 312225
7720  Ntot 280737
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7800  Ntot 273517
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8110  Ntot 356392
8111  Ntot 313956
8112  Ntot 327668
8113  Ntot 362065
8114  Ntot 361789
8115  Ntot 273227
8116  Ntot 370907
8117  Ntot 330192
8118  Ntot 304810
8119  Ntot 370375
8120  Ntot 290381
8121  Ntot 397245
8122  Ntot 276074
8123  Ntot 331320
8124  Ntot 407078
8125  Ntot 403951
8126  Ntot 367610
8127  Ntot 352627
8128  Ntot 344660
8129  Ntot 355003
8130  Ntot 353008
8131  Ntot 335935
8132  Ntot 341943
8133  Ntot 364740
8134  Ntot 350658
8135  Ntot 360988
8136  Ntot 369643
8137  Ntot 300051
8138  Ntot 357269
8139  Ntot 311753
8140  Ntot 297527
8141  Ntot 345266
8142  Ntot 360582
8143  Ntot 318776
8144  Ntot 339199
8145  Ntot 308376
8146  Ntot 382353
8147  Ntot 299790
8148  Ntot 325473
8149  Ntot 365035
8150  Ntot 309216
8151  Ntot 370183
8152  Ntot 303103
8153  Ntot 314795
8154  Ntot 449277
8155  Ntot 417233
8156  Ntot 383185
8157  Ntot 328327
8158  Ntot 391658
8159  Ntot 307503
8160  Ntot 350441
8161  Ntot 302057
8162  Ntot 364174
8163  Ntot 323792
8164  Ntot 356893
8165  Ntot 339283
8166  Ntot 336288
8167  Ntot 362146
8168  Ntot 421247
8169  Ntot 317410
8170  Ntot 326173
8171  Ntot 357470
8172  Ntot 350748
8173  Ntot 429904
8174  Ntot 357407
8175  Ntot 352501
8176  Ntot 340091
8177  Ntot 313068
8178  Ntot 297834
8179  Ntot 293237
8180  Ntot 298764
8181  Ntot 291752
8182  Ntot 287633
8183  Ntot 302883
8184  Ntot 376816
8185  Ntot 388284
8186  Ntot 318073
8187  Ntot 314180
8188  Ntot 318600
8189  Ntot 300366
8190  Ntot 318028
8191  Ntot 347609
8192  Ntot 364509
8193  Ntot 384582
8194  Ntot 337959
8195  Ntot 326226
8196  Ntot 312526
8197  Ntot 307955
8198  Ntot 330123
8199  Ntot 305489
8200  Ntot 350947
8201  Ntot 298518
8202  Ntot 344953
8203  Ntot 319636
8204  Ntot 340050
8205  Ntot 295838
8206  Ntot 363385
8207  Ntot 313636
8208  Ntot 314127
8209  Ntot 313946
8210  Ntot 314330
8211  Ntot 296924
8212  Ntot 306178
8213  Ntot 303729
8214  Ntot 329317
8215  Ntot 311289
8216  Ntot 362458
8217  Ntot 340384
8218  Ntot 363461
8219  Ntot 395084
8220  Ntot 362500
8221  Ntot 278534
8222  Ntot 332067
8223  Ntot 313993
8224  Ntot 295539
8225  Ntot 321528
8226  Ntot 331330
8227  Ntot 336376
8228  Ntot 302953
8229  Ntot 307534
8230  Ntot 307579
8231  Ntot 312512
8232  Ntot 305810
8233  Ntot 327635
8234  Ntot 323077
8235  Ntot 361517
8236  Ntot 311545
8237  Ntot 284240
8238  Ntot 291721
8239  Ntot 318373
8240  Ntot 347478
8241  Ntot 357295
8242  Ntot 362816
8243  Ntot 331443
8244  Ntot 330869
8245  Ntot 322124
8246  Ntot 397359
8247  Ntot 335610
8248  Ntot 341219
8249  Ntot 333115
8250  Ntot 307118
8251  Ntot 322058
8252  Ntot 361701
8253  Ntot 315377
8254  Ntot 356977
8255  Ntot 361787
8256  Ntot 302223
8257  Ntot 350403
8258  Ntot 337037
8259  Ntot 353771
8260  Ntot 307608
8261  Ntot 298071
8262  Ntot 316175
8263  Ntot 351998
8264  Ntot 330380
8265  Ntot 415626
8266  Ntot 302703
8267  Ntot 299042
8268  Ntot 390850
8269  Ntot 327220
8270  Ntot 319805
8271  Ntot 346518
8272  Ntot 323037
8273  Ntot 356319
8274  Ntot 293660
8275  Ntot 341846
8276  Ntot 385231
8277  Ntot 371349
8278  Ntot 325213
8279  Ntot 318439
8280  Ntot 291706
8281  Ntot 368877
8282  Ntot 321268
8283  Ntot 332597
8284  Ntot 340308
8285  Ntot 356764
8286  Ntot 368156
8287  Ntot 299901
8288  Ntot 322869
8289  Ntot 309895
8290  Ntot 286650
8291  Ntot 351783
8292  Ntot 288832
8293  Ntot 307057
8294  Ntot 303566
8295  Ntot 332396
8296  Ntot 421112
8297  Ntot 384955
8298  Ntot 363397
8299  Ntot 412766
8300  Ntot 308418
8301  Ntot 362824
8302  Ntot 363044
8303  Ntot 379404
8304  Ntot 344150
8305  Ntot 340272
8306  Ntot 321851
8307  Ntot 381337
8308  Ntot 309003
8309  Ntot 307263
8310  Ntot 332823
8311  Ntot 323950
8312  Ntot 329498
8313  Ntot 333544
8314  Ntot 422173
8315  Ntot 295348
8316  Ntot 357916
8317  Ntot 383821
8318  Ntot 367635
8319  Ntot 334425
8320  Ntot 339176
8321  Ntot 299709
8322  Ntot 346385
8323  Ntot 294075
8324  Ntot 362576
8325  Ntot 321503
8326  Ntot 393154
8327  Ntot 290852
8328  Ntot 345655
8329  Ntot 329447
8330  Ntot 379341
8331  Ntot 343408
8332  Ntot 361657
8333  Ntot 329640
8334  Ntot 411853
8335  Ntot 327365
8336  Ntot 337841
8337  Ntot 320800
8338  Ntot 304616
8339  Ntot 368859
8340  Ntot 297245
8341  Ntot 321873
8342  Ntot 359026
8343  Ntot 298970
8344  Ntot 356465
8345  Ntot 300669
8346  Ntot 313713
8347  Ntot 413106
8348  Ntot 349620
8349  Ntot 351410
8350  Ntot 301058
8351  Ntot 351194
8352  Ntot 350049
8353  Ntot 344028
8354  Ntot 307926
8355  Ntot 302288
8356  Ntot 330830
8357  Ntot 311241
8358  Ntot 343928
8359  Ntot 337917
8360  Ntot 357404
8361  Ntot 321890
8362  Ntot 321043
8363  Ntot 303221
8364  Ntot 345687
8365  Ntot 311765
8366  Ntot 396452
8367  Ntot 362074
8368  Ntot 354706
8369  Ntot 377533
8370  Ntot 316426
8371  Ntot 380794
8372  Ntot 312857
8373  Ntot 424794
8374  Ntot 481993
8375  Ntot 359068
8376  Ntot 316993
8377  Ntot 319638
8378  Ntot 307261
8379  Ntot 324996
8380  Ntot 305601
8381  Ntot 302276
8382  Ntot 288642
8383  Ntot 329584
8384  Ntot 326495
8385  Ntot 312529
8386  Ntot 339973
8387  Ntot 405091
8388  Ntot 313022
8389  Ntot 290744
8390  Ntot 325447
8391  Ntot 350040
8392  Ntot 305949
8393  Ntot 315460
8394  Ntot 282849
8395  Ntot 417798
8396  Ntot 355286
8397  Ntot 315579
8398  Ntot 306716
8399  Ntot 317879
8400  Ntot 345866
8401  Ntot 322140
8402  Ntot 285164
8403  Ntot 349039
8404  Ntot 374810
8405  Ntot 309632
8406  Ntot 313630
8407  Ntot 370679
8408  Ntot 333069
8409  Ntot 291254
8410  Ntot 304641
8411  Ntot 292196
8412  Ntot 321894
8413  Ntot 298423
8414  Ntot 356057
8415  Ntot 331333
8416  Ntot 286385
8417  Ntot 334727
8418  Ntot 305151
8419  Ntot 354016
8420  Ntot 312131
8421  Ntot 346417
8422  Ntot 317136
8423  Ntot 369462
8424  Ntot 325843
8425  Ntot 351892
8426  Ntot 300558
8427  Ntot 329756
8428  Ntot 289573
8429  Ntot 351369
8430  Ntot 315044
8431  Ntot 385545
8432  Ntot 327182
8433  Ntot 349804
8434  Ntot 337799
8435  Ntot 324715
8436  Ntot 323106
8437  Ntot 340868
8438  Ntot 382182
8439  Ntot 379422
8440  Ntot 309904
8441  Ntot 318029
8442  Ntot 304179
8443  Ntot 323319
8444  Ntot 311489
8445  Ntot 355082
8446  Ntot 294481
8447  Ntot 311206
8448  Ntot 297852
8449  Ntot 385867
8450  Ntot 366374
8451  Ntot 327509
8452  Ntot 340488
8453  Ntot 350995
8454  Ntot 307083
8455  Ntot 387558
8456  Ntot 336483
8457  Ntot 372893
8458  Ntot 359297
8459  Ntot 342828
8460  Ntot 332050
8461  Ntot 311213
8462  Ntot 325206
8463  Ntot 302722
8464  Ntot 328463
8465  Ntot 302061
8466  Ntot 330790
8467  Ntot 313240
8468  Ntot 365950
8469  Ntot 309122
8470  Ntot 335658
8471  Ntot 333921
8472  Ntot 308484
8473  Ntot 360713
8474  Ntot 316758
8475  Ntot 372514
8476  Ntot 353664
8477  Ntot 379911
8478  Ntot 372671
8479  Ntot 325564
8480  Ntot 409437
8481  Ntot 337470
8482  Ntot 361899
8483  Ntot 327207
8484  Ntot 348047
8485  Ntot 338736
8486  Ntot 373115
8487  Ntot 315737
8488  Ntot 354901
8489  Ntot 377747
8490  Ntot 386805
8491  Ntot 312033
8492  Ntot 354788
8493  Ntot 348771
8494  Ntot 316766
8495  Ntot 304996
8496  Ntot 320112
8497  Ntot 438132
8498  Ntot 343214
8499  Ntot 346131
8500  Ntot 339383
8501  Ntot 332541
8502  Ntot 359881
8503  Ntot 338432
8504  Ntot 291124
8505  Ntot 336739
8506  Ntot 311297
8507  Ntot 365281
8508  Ntot 320901
8509  Ntot 367497
8510  Ntot 315477
8511  Ntot 337203
8512  Ntot 362645
8513  Ntot 313583
8514  Ntot 331726
8515  Ntot 338715
8516  Ntot 326842
8517  Ntot 351465
8518  Ntot 372909
8519  Ntot 285486
8520  Ntot 314917
8521  Ntot 311864
8522  Ntot 402043
8523  Ntot 342189
8524  Ntot 337567
8525  Ntot 326320
8526  Ntot 340795
8527  Ntot 361202
8528  Ntot 334229
8529  Ntot 305417
8530  Ntot 328811
8531  Ntot 351040
8532  Ntot 351435
8533  Ntot 324114
8534  Ntot 357777
8535  Ntot 377733
8536  Ntot 302479
8537  Ntot 366745
8538  Ntot 319733
8539  Ntot 288429
8540  Ntot 347125
8541  Ntot 364791
8542  Ntot 321501
8543  Ntot 349631
8544  Ntot 307839
8545  Ntot 283840
8546  Ntot 403899
8547  Ntot 347016
8548  Ntot 318419
8549  Ntot 395186
8550  Ntot 385482
8551  Ntot 343160
8552  Ntot 316849
8553  Ntot 360521
8554  Ntot 362893
8555  Ntot 304261
8556  Ntot 315251
8557  Ntot 351859
8558  Ntot 363128
8559  Ntot 303324
8560  Ntot 284821
8561  Ntot 387054
8562  Ntot 382175
8563  Ntot 348362
8564  Ntot 287638
8565  Ntot 336972
8566  Ntot 351514
8567  Ntot 278801
8568  Ntot 329758
8569  Ntot 274989
8570  Ntot 454423
8571  Ntot 354588
8572  Ntot 333894
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8574  Ntot 386165
8575  Ntot 369478
8576  Ntot 336469
8577  Ntot 357881
8578  Ntot 327176
8579  Ntot 342406
8580  Ntot 404133
8581  Ntot 309056
8582  Ntot 316491
8583  Ntot 391926
8584  Ntot 318894
8585  Ntot 332372
8586  Ntot 358069
8587  Ntot 317392
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8589  Ntot 375097
8590  Ntot 348839
8591  Ntot 353984
8592  Ntot 322580
8593  Ntot 339554
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8595  Ntot 311545
8596  Ntot 310963
8597  Ntot 333905
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8599  Ntot 293819
8600  Ntot 295569
8601  Ntot 343757
8602  Ntot 309632
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8604  Ntot 328307
8605  Ntot 315726
8606  Ntot 433790
8607  Ntot 335448
8608  Ntot 334973
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8610  Ntot 305409
8611  Ntot 338595
8612  Ntot 345206
8613  Ntot 344425
8614  Ntot 301752
8615  Ntot 298397
8616  Ntot 325198
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8619  Ntot 370827
8620  Ntot 345567
8621  Ntot 343494
8622  Ntot 369797
8623  Ntot 316500
8624  Ntot 333446
8625  Ntot 328261
8626  Ntot 351503
8627  Ntot 342547
8628  Ntot 335257
8629  Ntot 305694
8630  Ntot 326801
8631  Ntot 377057
8632  Ntot 296894
8633  Ntot 365282
8634  Ntot 338032
8635  Ntot 313207
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8637  Ntot 356174
8638  Ntot 463483
8639  Ntot 352828
8640  Ntot 333441
8641  Ntot 327549
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8644  Ntot 375399
8645  Ntot 330254
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8647  Ntot 360479
8648  Ntot 282426
8649  Ntot 301092
8650  Ntot 409949
8651  Ntot 343355
8652  Ntot 369605
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8654  Ntot 375931
8655  Ntot 340490
8656  Ntot 297342
8657  Ntot 364682
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8660  Ntot 424119
8661  Ntot 310997
8662  Ntot 341200
8663  Ntot 339494
8664  Ntot 311718
8665  Ntot 313405
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8667  Ntot 357043
8668  Ntot 305458
8669  Ntot 391049
8670  Ntot 316053
8671  Ntot 412197
8672  Ntot 337594
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8674  Ntot 318866
8675  Ntot 292105
8676  Ntot 439990
8677  Ntot 372653
8678  Ntot 322497
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8680  Ntot 454179
8681  Ntot 317805
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8684  Ntot 332971
8685  Ntot 355023
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8688  Ntot 401733
8689  Ntot 356698
8690  Ntot 371018
8691  Ntot 328666
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8693  Ntot 352714
8694  Ntot 340026
8695  Ntot 390375
8696  Ntot 395944
8697  Ntot 401233
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8699  Ntot 365261
8700  Ntot 308273
8701  Ntot 311318
8702  Ntot 336520
8703  Ntot 326896
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8705  Ntot 301441
8706  Ntot 336984
8707  Ntot 319996
8708  Ntot 301822
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8710  Ntot 295359
8711  Ntot 373870
8712  Ntot 303611
8713  Ntot 312474
8714  Ntot 370997
8715  Ntot 364528
8716  Ntot 322627
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8718  Ntot 357264
8719  Ntot 339753
8720  Ntot 306788
8721  Ntot 326882
8722  Ntot 337381
8723  Ntot 317861
8724  Ntot 313086
8725  Ntot 402992
8726  Ntot 329710
8727  Ntot 321881
8728  Ntot 394778
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8730  Ntot 323545
8731  Ntot 341130
8732  Ntot 368257
8733  Ntot 328480
8734  Ntot 382534
8735  Ntot 319592
8736  Ntot 432957
8737  Ntot 407334
8738  Ntot 308294
8739  Ntot 389809
8740  Ntot 310195
8741  Ntot 394729
8742  Ntot 389421
8743  Ntot 336335
8744  Ntot 313554
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8746  Ntot 302388
8747  Ntot 297704
8748  Ntot 428595
8749  Ntot 323516
8750  Ntot 383969
8751  Ntot 341585
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8753  Ntot 329808
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8755  Ntot 306995
8756  Ntot 336692
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8758  Ntot 334996
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8760  Ntot 329846
8761  Ntot 305889
8762  Ntot 289544
8763  Ntot 312944
8764  Ntot 306210
8765  Ntot 325802
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8767  Ntot 305861
8768  Ntot 318146
8769  Ntot 392192
8770  Ntot 358615
8771  Ntot 320896
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8773  Ntot 279515
8774  Ntot 418843
8775  Ntot 452942
8776  Ntot 335143
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8778  Ntot 311437
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8780  Ntot 347591
8781  Ntot 299186
8782  Ntot 364461
8783  Ntot 323039
8784  Ntot 347667
8785  Ntot 273366
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8787  Ntot 311713
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8789  Ntot 408582
8790  Ntot 386318
8791  Ntot 341373
8792  Ntot 312415
8793  Ntot 296321
8794  Ntot 284001
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8796  Ntot 330008
8797  Ntot 400499
8798  Ntot 369214
8799  Ntot 333312
8800  Ntot 328985
8801  Ntot 412606
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8810  Ntot 328372
8811  Ntot 402293
8812  Ntot 333957
8813  Ntot 296188
8814  Ntot 407606
8815  Ntot 353407
8816  Ntot 312558
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8818  Ntot 425086
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8820  Ntot 339266
8821  Ntot 316999
8822  Ntot 336392
8823  Ntot 309551
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8825  Ntot 331031
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8827  Ntot 332410
8828  Ntot 300050
8829  Ntot 305230
8830  Ntot 346573
8831  Ntot 362391
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8833  Ntot 327251
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8838  Ntot 312152
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8844  Ntot 313337
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8847  Ntot 385189
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8859  Ntot 297786
8860  Ntot 394857
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8900  Ntot 296237
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8904  Ntot 352947
8905  Ntot 294695
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8910  Ntot 325690
8911  Ntot 387415
8912  Ntot 337455
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8961  Ntot 398131
8962  Ntot 343829
8963  Ntot 353889
8964  Ntot 326225
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8966  Ntot 302435
8967  Ntot 376146
8968  Ntot 324476
8969  Ntot 327486
8970  Ntot 359169
8971  Ntot 287575
8972  Ntot 361217
8973  Ntot 307387
8974  Ntot 314419
8975  Ntot 370089
8976  Ntot 296109
8977  Ntot 295983
8978  Ntot 318189
8979  Ntot 358978
8980  Ntot 359053
8981  Ntot 352215
8982  Ntot 309025
8983  Ntot 310213
8984  Ntot 382236
8985  Ntot 339867
8986  Ntot 329443
8987  Ntot 306764
8988  Ntot 377221
8989  Ntot 309052
8990  Ntot 287277
8991  Ntot 371446
8992  Ntot 318940
8993  Ntot 301242
8994  Ntot 346621
8995  Ntot 303649
8996  Ntot 311266
8997  Ntot 331239
8998  Ntot 313042
8999  Ntot 308248
9000  Ntot 369164
9001  Ntot 361941
9002  Ntot 285231
9003  Ntot 383632
9004  Ntot 350786
9005  Ntot 288146
9006  Ntot 331759
9007  Ntot 357463
9008  Ntot 303381
9009  Ntot 364158
9010  Ntot 386109
9011  Ntot 314793
9012  Ntot 353331
9013  Ntot 334068
9014  Ntot 302123
9015  Ntot 303791
9016  Ntot 355097
9017  Ntot 378696
9018  Ntot 297201
9019  Ntot 355626
9020  Ntot 282424
9021  Ntot 344237
9022  Ntot 381372
9023  Ntot 321802
9024  Ntot 313562
9025  Ntot 330718
9026  Ntot 381787
9027  Ntot 282212
9028  Ntot 346594
9029  Ntot 353786
9030  Ntot 357893
9031  Ntot 374691
9032  Ntot 332527
9033  Ntot 332746
9034  Ntot 390434
9035  Ntot 386981
9036  Ntot 323744
9037  Ntot 342818
9038  Ntot 339140
9039  Ntot 346437
9040  Ntot 307139
9041  Ntot 316651
9042  Ntot 381853
9043  Ntot 337789
9044  Ntot 405911
9045  Ntot 336780
9046  Ntot 356793
9047  Ntot 364121
9048  Ntot 329656
9049  Ntot 314856
9050  Ntot 548006
9051  Ntot 331475
9052  Ntot 309271
9053  Ntot 344968
9054  Ntot 280090
9055  Ntot 410117
9056  Ntot 358043
9057  Ntot 297682
9058  Ntot 308096
9059  Ntot 335961
9060  Ntot 293518
9061  Ntot 358090
9062  Ntot 354459
9063  Ntot 291637
9064  Ntot 345143
9065  Ntot 322355
9066  Ntot 331883
9067  Ntot 345430
9068  Ntot 340936
9069  Ntot 371595
9070  Ntot 307667
9071  Ntot 313354
9072  Ntot 328068
9073  Ntot 324429
9074  Ntot 330006
9075  Ntot 319063
9076  Ntot 427773
9077  Ntot 366833
9078  Ntot 325780
9079  Ntot 310023
9080  Ntot 336255
9081  Ntot 367288
9082  Ntot 318311
9083  Ntot 318778
9084  Ntot 335180
9085  Ntot 348803
9086  Ntot 297992
9087  Ntot 315388
9088  Ntot 346594
9089  Ntot 315468
9090  Ntot 349178
9091  Ntot 328682
9092  Ntot 339515
9093  Ntot 343795
9094  Ntot 346594
9095  Ntot 388539
9096  Ntot 344913
9097  Ntot 368252
9098  Ntot 353701
9099  Ntot 367709
9100  Ntot 401326
9101  Ntot 332467
9102  Ntot 339330
9103  Ntot 342934
9104  Ntot 342403
9105  Ntot 337534
9106  Ntot 318465
9107  Ntot 381669
9108  Ntot 353601
9109  Ntot 303573
9110  Ntot 342692
9111  Ntot 332330
9112  Ntot 327946
9113  Ntot 329598
9114  Ntot 317908
9115  Ntot 351841
9116  Ntot 306804
9117  Ntot 413121
9118  Ntot 321035
9119  Ntot 356317
9120  Ntot 310680
9121  Ntot 276217
9122  Ntot 326570
9123  Ntot 349135
9124  Ntot 354917
9125  Ntot 363656
9126  Ntot 384816
9127  Ntot 313385
9128  Ntot 321503
9129  Ntot 346982
9130  Ntot 337704
9131  Ntot 335052
9132  Ntot 302693
9133  Ntot 356074
9134  Ntot 321988
9135  Ntot 296008
9136  Ntot 351637
9137  Ntot 367945
9138  Ntot 312242
9139  Ntot 360102
9140  Ntot 338985
9141  Ntot 409077
9142  Ntot 292113
9143  Ntot 309166
9144  Ntot 358038
9145  Ntot 390252
9146  Ntot 342522
9147  Ntot 345107
9148  Ntot 352737
9149  Ntot 412001
9150  Ntot 361695
9151  Ntot 288535
9152  Ntot 309973
9153  Ntot 317967
9154  Ntot 379019
9155  Ntot 331154
9156  Ntot 293736
9157  Ntot 318878
9158  Ntot 373674
9159  Ntot 285272
9160  Ntot 340970
9161  Ntot 438437
9162  Ntot 338325
9163  Ntot 370629
9164  Ntot 387931
9165  Ntot 329590
9166  Ntot 310523
9167  Ntot 308447
9168  Ntot 383624
9169  Ntot 334852
9170  Ntot 327206
9171  Ntot 371984
9172  Ntot 279201
9173  Ntot 353520
9174  Ntot 284934
9175  Ntot 321214
9176  Ntot 374538
9177  Ntot 342057
9178  Ntot 325104
9179  Ntot 314703
9180  Ntot 332421
9181  Ntot 379580
9182  Ntot 315385
9183  Ntot 315263
9184  Ntot 363790
9185  Ntot 315917
9186  Ntot 355950
9187  Ntot 299922
9188  Ntot 296864
9189  Ntot 385268
9190  Ntot 344429
9191  Ntot 316868
9192  Ntot 305046
9193  Ntot 330464
9194  Ntot 287448
9195  Ntot 298562
9196  Ntot 331302
9197  Ntot 301846
9198  Ntot 275431
9199  Ntot 330217
9200  Ntot 326657
9201  Ntot 365095
9202  Ntot 341237
9203  Ntot 318844
9204  Ntot 345378
9205  Ntot 326099
9206  Ntot 293849
9207  Ntot 360418
9208  Ntot 371706
9209  Ntot 304814
9210  Ntot 339891
9211  Ntot 349100
9212  Ntot 409177
9213  Ntot 290649
9214  Ntot 416251
9215  Ntot 332677
9216  Ntot 355642
9217  Ntot 331538
9218  Ntot 327119
9219  Ntot 332886
9220  Ntot 347503
9221  Ntot 322316
9222  Ntot 349315
9223  Ntot 344890
9224  Ntot 353278
9225  Ntot 457376
9226  Ntot 321134
9227  Ntot 314180
9228  Ntot 315739
9229  Ntot 354093
9230  Ntot 331105
9231  Ntot 287901
9232  Ntot 340073
9233  Ntot 337375
9234  Ntot 316669
9235  Ntot 345306
9236  Ntot 311657
9237  Ntot 309174
9238  Ntot 315213
9239  Ntot 301827
9240  Ntot 326275
9241  Ntot 343073
9242  Ntot 513142
9243  Ntot 325640
9244  Ntot 299104
9245  Ntot 313423
9246  Ntot 366091
9247  Ntot 488900
9248  Ntot 291732
9249  Ntot 351378
9250  Ntot 339830
9251  Ntot 295104
9252  Ntot 326749
9253  Ntot 335827
9254  Ntot 417230
9255  Ntot 373521
9256  Ntot 359239
9257  Ntot 324510
9258  Ntot 293373
9259  Ntot 309126
9260  Ntot 315198
9261  Ntot 350582
9262  Ntot 305104
9263  Ntot 352703
9264  Ntot 296016
9265  Ntot 326062
9266  Ntot 329465
9267  Ntot 371288
9268  Ntot 323745
9269  Ntot 344310
9270  Ntot 356861
9271  Ntot 303015
9272  Ntot 346333
9273  Ntot 307028
9274  Ntot 301582
9275  Ntot 327816
9276  Ntot 310925
9277  Ntot 295086
9278  Ntot 325374
9279  Ntot 296526
9280  Ntot 355857
9281  Ntot 296626
9282  Ntot 324684
9283  Ntot 355214
9284  Ntot 336639
9285  Ntot 360338
9286  Ntot 349676
9287  Ntot 398707
9288  Ntot 321530
9289  Ntot 310645
9290  Ntot 326504
9291  Ntot 312382
9292  Ntot 320493
9293  Ntot 322442
9294  Ntot 330985
9295  Ntot 335396
9296  Ntot 332059
9297  Ntot 328108
9298  Ntot 305717
9299  Ntot 313798
9300  Ntot 287743
9301  Ntot 349079
9302  Ntot 311139
9303  Ntot 322639
9304  Ntot 330970
9305  Ntot 383239
9306  Ntot 336904
9307  Ntot 341792
9308  Ntot 420493
9309  Ntot 315370
9310  Ntot 303049
9311  Ntot 353895
9312  Ntot 324183
9313  Ntot 337750
9314  Ntot 322401
9315  Ntot 287641
9316  Ntot 371904
9317  Ntot 328958
9318  Ntot 316639
9319  Ntot 373445
9320  Ntot 360104
9321  Ntot 339732
9322  Ntot 306407
9323  Ntot 316473
9324  Ntot 351800
9325  Ntot 280041
9326  Ntot 345641
9327  Ntot 311357
9328  Ntot 361622
9329  Ntot 343492
9330  Ntot 325544
9331  Ntot 348904
9332  Ntot 308507
9333  Ntot 363946
9334  Ntot 333645
9335  Ntot 341794
9336  Ntot 334957
9337  Ntot 306706
9338  Ntot 357115
9339  Ntot 352735
9340  Ntot 403365
9341  Ntot 317450
9342  Ntot 398106
9343  Ntot 318404
9344  Ntot 350120
9345  Ntot 341555
9346  Ntot 283206
9347  Ntot 391904
9348  Ntot 356574
9349  Ntot 306930
9350  Ntot 369634
9351  Ntot 377611
9352  Ntot 301244
9353  Ntot 276016
9354  Ntot 365079
9355  Ntot 337165
9356  Ntot 333152
9357  Ntot 362253
9358  Ntot 313808
9359  Ntot 346553
9360  Ntot 351314
9361  Ntot 320838
9362  Ntot 297779
9363  Ntot 319127
9364  Ntot 354124
9365  Ntot 307327
9366  Ntot 321774
9367  Ntot 317622
9368  Ntot 333633
9369  Ntot 324672
9370  Ntot 309505
9371  Ntot 324238
9372  Ntot 338111
9373  Ntot 346053
9374  Ntot 333948
9375  Ntot 323995
9376  Ntot 342272
9377  Ntot 329290
9378  Ntot 305436
9379  Ntot 390581
9380  Ntot 386914
9381  Ntot 346476
9382  Ntot 410333
9383  Ntot 305927
9384  Ntot 327086
9385  Ntot 351679
9386  Ntot 348256
9387  Ntot 324190
9388  Ntot 374195
9389  Ntot 327274
9390  Ntot 310258
9391  Ntot 323690
9392  Ntot 328533
9393  Ntot 295843
9394  Ntot 372764
9395  Ntot 327586
9396  Ntot 338660
9397  Ntot 279172
9398  Ntot 426997
9399  Ntot 286569
9400  Ntot 334362
9401  Ntot 345592
9402  Ntot 329020
9403  Ntot 371326
9404  Ntot 404123
9405  Ntot 318905
9406  Ntot 366608
9407  Ntot 323639
9408  Ntot 314136
9409  Ntot 340984
9410  Ntot 310120
9411  Ntot 342341
9412  Ntot 331455
9413  Ntot 310928
9414  Ntot 345109
9415  Ntot 289691
9416  Ntot 306354
9417  Ntot 323412
9418  Ntot 378734
9419  Ntot 325938
9420  Ntot 340779
9421  Ntot 322007
9422  Ntot 336696
9423  Ntot 347614
9424  Ntot 335437
9425  Ntot 312689
9426  Ntot 296564
9427  Ntot 331162
9428  Ntot 293491
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9430  Ntot 333197
9431  Ntot 309596
9432  Ntot 399981
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9434  Ntot 303008
9435  Ntot 386615
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9437  Ntot 309127
9438  Ntot 367429
9439  Ntot 368265
9440  Ntot 300988
9441  Ntot 342795
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9443  Ntot 333641
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9445  Ntot 326025
9446  Ntot 313358
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9449  Ntot 314225
9450  Ntot 333483
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9452  Ntot 337450
9453  Ntot 381378
9454  Ntot 290028
9455  Ntot 356456
9456  Ntot 330909
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9458  Ntot 447267
9459  Ntot 324658
9460  Ntot 297846
9461  Ntot 315232
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9464  Ntot 402455
9465  Ntot 402850
9466  Ntot 349838
9467  Ntot 301318
9468  Ntot 336424
9469  Ntot 288111
9470  Ntot 322185
9471  Ntot 399572
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9473  Ntot 337967
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9475  Ntot 322418
9476  Ntot 335856
9477  Ntot 362789
9478  Ntot 289138
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9480  Ntot 366238
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9482  Ntot 323141
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9485  Ntot 363115
9486  Ntot 291841
9487  Ntot 350054
9488  Ntot 305291
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9490  Ntot 306603
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9492  Ntot 354425
9493  Ntot 319828
9494  Ntot 321891
9495  Ntot 353983
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9499  Ntot 296657
9500  Ntot 365752
9501  Ntot 349253
9502  Ntot 357280
9503  Ntot 382353
9504  Ntot 353622
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9506  Ntot 311593
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9510  Ntot 312059
9511  Ntot 403028
9512  Ntot 394593
9513  Ntot 364259
9514  Ntot 345886
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9516  Ntot 282450
9517  Ntot 313320
9518  Ntot 305283
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9520  Ntot 363210
9521  Ntot 311915
9522  Ntot 326770
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9524  Ntot 386543
9525  Ntot 369413
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9527  Ntot 318393
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9529  Ntot 337087
9530  Ntot 313856
9531  Ntot 321733
9532  Ntot 370402
9533  Ntot 325006
9534  Ntot 386484
9535  Ntot 340853
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9537  Ntot 310891
9538  Ntot 320176
9539  Ntot 373195
9540  Ntot 390393
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9542  Ntot 336738
9543  Ntot 297471
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9545  Ntot 445182
9546  Ntot 374575
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9548  Ntot 295714
9549  Ntot 302753
9550  Ntot 354826
9551  Ntot 339171
9552  Ntot 315923
9553  Ntot 327523
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9555  Ntot 317493
9556  Ntot 321573
9557  Ntot 372002
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9559  Ntot 291084
9560  Ntot 371628
9561  Ntot 368789
9562  Ntot 407592
9563  Ntot 317731
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9565  Ntot 403481
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9567  Ntot 394134
9568  Ntot 348887
9569  Ntot 324639
9570  Ntot 341556
9571  Ntot 360221
9572  Ntot 311621
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9575  Ntot 375926
9576  Ntot 298567
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9578  Ntot 333363
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9580  Ntot 331990
9581  Ntot 332986
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9583  Ntot 328668
9584  Ntot 388865
9585  Ntot 324459
9586  Ntot 328281
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9588  Ntot 319740
9589  Ntot 293282
9590  Ntot 329820
9591  Ntot 396669
9592  Ntot 313273
9593  Ntot 275853
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9595  Ntot 318868
9596  Ntot 305450
9597  Ntot 304459
9598  Ntot 432358
9599  Ntot 349911
9600  Ntot 315515
9601  Ntot 407445
9602  Ntot 421565
9603  Ntot 306562
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9606  Ntot 316205
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9610  Ntot 351450
9611  Ntot 303676
9612  Ntot 323149
9613  Ntot 307398
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9615  Ntot 360487
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9618  Ntot 350846
9619  Ntot 322727
9620  Ntot 373883
9621  Ntot 310897
9622  Ntot 342452
9623  Ntot 312856
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9625  Ntot 320232
9626  Ntot 411104
9627  Ntot 297585
9628  Ntot 320012
9629  Ntot 294794
9630  Ntot 350596
9631  Ntot 295441
9632  Ntot 353791
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9634  Ntot 408818
9635  Ntot 359521
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9637  Ntot 405759
9638  Ntot 317539
9639  Ntot 316691
9640  Ntot 357903
9641  Ntot 329001
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9650  Ntot 364932
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9660  Ntot 315238
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9664  Ntot 296410
9665  Ntot 312401
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9667  Ntot 312785
9668  Ntot 350501
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9670  Ntot 325246
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9711  Ntot 328379
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9722  Ntot 418012
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9781  Ntot 341542
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9801  Ntot 369392
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9814  Ntot 334592
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9819  Ntot 321269
9820  Ntot 335334
9821  Ntot 352484
9822  Ntot 346013
9823  Ntot 314005
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9830  Ntot 340800
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9836  Ntot 369041
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9839  Ntot 292480
9840  Ntot 361761
9841  Ntot 343015
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9849  Ntot 335952
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9860  Ntot 299367
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9864  Ntot 431861
9865  Ntot 350876
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9867  Ntot 307753
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9875  Ntot 297518
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9877  Ntot 359415
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9880  Ntot 396993
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9890  Ntot 348339
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9893  Ntot 325751
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9895  Ntot 329551
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9899  Ntot 377281
9900  Ntot 359660
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9902  Ntot 337237
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9906  Ntot 382863
9907  Ntot 283900
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9955  Ntot 324899
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9957  Ntot 308968
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9978  Ntot 300591
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9983  Ntot 375453
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9995  Ntot 308546
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9999  Ntot 361604
10000 Ntot 326508
10001 Ntot 315026
10002 Ntot 318097
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10004 Ntot 333305
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10765 Ntot 338340
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10768 Ntot 318014
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10775 Ntot 267491
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10780 Ntot 291680
10781 Ntot 376103
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10784 Ntot 324744
10785 Ntot 384198
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10788 Ntot 317383
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10790 Ntot 315452
10791 Ntot 393145
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10793 Ntot 323761
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10795 Ntot 353887
10796 Ntot 358673
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10799 Ntot 321544
10800 Ntot 282383
10801 Ntot 345009
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10805 Ntot 325906
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10808 Ntot 333567
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10810 Ntot 339901
10811 Ntot 312221
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10813 Ntot 427518
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10822 Ntot 274064
10823 Ntot 365722
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10825 Ntot 328606
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10827 Ntot 334193
10828 Ntot 342574
10829 Ntot 323206
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10832 Ntot 386551
10833 Ntot 386797
10834 Ntot 330057
10835 Ntot 314853
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10837 Ntot 321954
10838 Ntot 420558
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10840 Ntot 316496
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10844 Ntot 293309
10845 Ntot 302965
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10847 Ntot 344687
10848 Ntot 337317
10849 Ntot 314817
10850 Ntot 314903
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10860 Ntot 304358
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10865 Ntot 350176
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10867 Ntot 319955
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10873 Ntot 360372
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10875 Ntot 317848
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10877 Ntot 329306
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10895 Ntot 313032
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10899 Ntot 300740
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10904 Ntot 350198
10905 Ntot 435573
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10910 Ntot 320528
10911 Ntot 312843
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10924 Ntot 404370
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10929 Ntot 390875
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10931 Ntot 348109
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10933 Ntot 304767
10934 Ntot 374851
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10937 Ntot 291667
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10946 Ntot 299429
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10962 Ntot 305242
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11717 Ntot 344933
11718 Ntot 301080
11719 Ntot 320836
11720 Ntot 320994
11721 Ntot 320298
11722 Ntot 304091
11723 Ntot 328273
11724 Ntot 336829
11725 Ntot 341627
11726 Ntot 318513
11727 Ntot 306157
11728 Ntot 333627
11729 Ntot 342561
11730 Ntot 344658
11731 Ntot 313169
11732 Ntot 318924
11733 Ntot 377902
11734 Ntot 327358
11735 Ntot 335253
11736 Ntot 330266
11737 Ntot 349891
11738 Ntot 361899
11739 Ntot 316913
11740 Ntot 373321
11741 Ntot 310285
11742 Ntot 335672
11743 Ntot 289097
11744 Ntot 331677
11745 Ntot 336277
11746 Ntot 284142
11747 Ntot 325292
11748 Ntot 393526
11749 Ntot 320630
11750 Ntot 308826
11751 Ntot 327830
11752 Ntot 309778
11753 Ntot 309043
11754 Ntot 415623
11755 Ntot 317398
11756 Ntot 297713
11757 Ntot 313862
11758 Ntot 349173
11759 Ntot 308788
11760 Ntot 341801
11761 Ntot 341407
11762 Ntot 350843
11763 Ntot 379279
11764 Ntot 358101
11765 Ntot 337332
11766 Ntot 395908
11767 Ntot 314105
11768 Ntot 313075
11769 Ntot 311341
11770 Ntot 311376
11771 Ntot 316591
11772 Ntot 341059
11773 Ntot 331854
11774 Ntot 291183
11775 Ntot 388230
11776 Ntot 350633
11777 Ntot 333892
11778 Ntot 306055
11779 Ntot 415032
11780 Ntot 307572
11781 Ntot 436631
11782 Ntot 319912
11783 Ntot 306476
11784 Ntot 329015
11785 Ntot 314253
11786 Ntot 333195
11787 Ntot 286129
11788 Ntot 365219
11789 Ntot 334585
11790 Ntot 299253
11791 Ntot 308494
11792 Ntot 324204
11793 Ntot 323074
11794 Ntot 322020
11795 Ntot 288048
11796 Ntot 400346
11797 Ntot 312640
11798 Ntot 305009
11799 Ntot 329862
11800 Ntot 374276
11801 Ntot 382082
11802 Ntot 369622
11803 Ntot 346420
11804 Ntot 369084
11805 Ntot 274239
11806 Ntot 300776
11807 Ntot 274050
11808 Ntot 337758
11809 Ntot 334418
11810 Ntot 320990
11811 Ntot 360624
11812 Ntot 381545
11813 Ntot 320886
11814 Ntot 387702
11815 Ntot 449034
11816 Ntot 323967
11817 Ntot 370059
11818 Ntot 334706
11819 Ntot 290953
11820 Ntot 372667
11821 Ntot 294919
11822 Ntot 393851
11823 Ntot 380667
11824 Ntot 303858
11825 Ntot 381577
11826 Ntot 443164
11827 Ntot 362624
11828 Ntot 339146
11829 Ntot 328307
11830 Ntot 346492
11831 Ntot 354611
11832 Ntot 389428
11833 Ntot 307098
11834 Ntot 310324
11835 Ntot 280535
11836 Ntot 334388
11837 Ntot 470871
11838 Ntot 346404
11839 Ntot 322977
11840 Ntot 297598
11841 Ntot 311258
11842 Ntot 338518
11843 Ntot 396130
11844 Ntot 465355
11845 Ntot 320240
11846 Ntot 323228
11847 Ntot 370780
11848 Ntot 314672
11849 Ntot 339173
11850 Ntot 313191
11851 Ntot 433138
11852 Ntot 329351
11853 Ntot 317622
11854 Ntot 301365
11855 Ntot 423550
11856 Ntot 315497
11857 Ntot 353537
11858 Ntot 302679
11859 Ntot 334037
11860 Ntot 311565
11861 Ntot 308359
11862 Ntot 309008
11863 Ntot 312329
11864 Ntot 336308
11865 Ntot 373397
11866 Ntot 296566
11867 Ntot 319942
11868 Ntot 279078
11869 Ntot 338944
11870 Ntot 341581
11871 Ntot 346033
11872 Ntot 303784
11873 Ntot 331792
11874 Ntot 322883
11875 Ntot 343114
11876 Ntot 287720
11877 Ntot 313034
11878 Ntot 319091
11879 Ntot 335835
11880 Ntot 305378
11881 Ntot 356550
11882 Ntot 326163
11883 Ntot 319242
11884 Ntot 328769
11885 Ntot 325298
11886 Ntot 323114
11887 Ntot 282513
11888 Ntot 298705
11889 Ntot 304251
11890 Ntot 335926
11891 Ntot 344133
11892 Ntot 357366
11893 Ntot 340781
11894 Ntot 334375
11895 Ntot 309908
11896 Ntot 375085
11897 Ntot 323848
11898 Ntot 339379
11899 Ntot 361603
11900 Ntot 293011
11901 Ntot 312153
11902 Ntot 345700
11903 Ntot 398054
11904 Ntot 323091
11905 Ntot 345871
11906 Ntot 319348
11907 Ntot 293504
11908 Ntot 296351
11909 Ntot 357856
11910 Ntot 330401
11911 Ntot 327461
11912 Ntot 376307
11913 Ntot 423748
11914 Ntot 285129
11915 Ntot 318895
11916 Ntot 308235
11917 Ntot 346445
11918 Ntot 296007
11919 Ntot 343445
11920 Ntot 317158
11921 Ntot 332656
11922 Ntot 376968
11923 Ntot 316583
11924 Ntot 359496
11925 Ntot 315976
11926 Ntot 308634
11927 Ntot 366991
11928 Ntot 347847
11929 Ntot 386467
11930 Ntot 339840
11931 Ntot 352696
11932 Ntot 369304
11933 Ntot 387867
11934 Ntot 375623
11935 Ntot 411407
11936 Ntot 340580
11937 Ntot 310877
11938 Ntot 387150
11939 Ntot 373582
11940 Ntot 334730
11941 Ntot 389567
11942 Ntot 359516
11943 Ntot 356580
11944 Ntot 285121
11945 Ntot 362913
11946 Ntot 338992
11947 Ntot 291693
11948 Ntot 317458
11949 Ntot 344434
11950 Ntot 380422
11951 Ntot 292495
11952 Ntot 306634
11953 Ntot 278286
11954 Ntot 292538
11955 Ntot 317466
11956 Ntot 331676
11957 Ntot 399278
11958 Ntot 298156
11959 Ntot 294273
11960 Ntot 283873
11961 Ntot 348623
11962 Ntot 372724
11963 Ntot 327969
11964 Ntot 364609
11965 Ntot 305753
11966 Ntot 314148
11967 Ntot 321530
11968 Ntot 328439
11969 Ntot 371368
11970 Ntot 361458
11971 Ntot 353400
11972 Ntot 339945
11973 Ntot 325975
11974 Ntot 336875
11975 Ntot 363753
11976 Ntot 356750
11977 Ntot 328727
11978 Ntot 338878
11979 Ntot 350613
11980 Ntot 387923
11981 Ntot 327024
11982 Ntot 340359
11983 Ntot 384141
11984 Ntot 347789
11985 Ntot 327330
11986 Ntot 345045
11987 Ntot 331190
11988 Ntot 337261
11989 Ntot 348224
11990 Ntot 329237
11991 Ntot 366574
11992 Ntot 331781
11993 Ntot 318644
11994 Ntot 280169
11995 Ntot 314630
11996 Ntot 334866
11997 Ntot 319733
11998 Ntot 350535
11999 Ntot 386078
12000 Ntot 340368

$layers
$layers[[1]]
mapping: xintercept = ~quants 
geom_vline: na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 

$layers[[2]]
geom_density: na.rm = FALSE, orientation = NA, outline.type = upper
stat_density: na.rm = FALSE, orientation = NA
position_identity 

$layers[[3]]
mapping: label = ~paste("Posterior mean: ", post.mean, "\nPosterior sd  : ", post.sd, sep = ""), vjust = 1, hjust = 1, x = Inf, y = Inf 
geom_text: parse = FALSE, check_overlap = FALSE, size.unit = mm, na.rm = FALSE
stat_identity: na.rm = FALSE
position_identity 


$scales
<ggproto object: Class ScalesList, gg>
    add: function
    add_defaults: function
    add_missing: function
    backtransform_df: function
    clone: function
    find: function
    get_scales: function
    has_scale: function
    input: function
    map_df: function
    n: function
    non_position_scales: function
    scales: list
    set_palettes: function
    train_df: function
    transform_df: function
    super:  <ggproto object: Class ScalesList, gg>

$guides
<Guides[0] ggproto object>

<empty>

$mapping
$x
<quosure>
expr: ^sample
env:  0x7fc5d3ddc570

$y
<quosure>
expr: ^..density..
env:  0x7fc5d3ddc570

attr(,"class")
[1] "uneval"

$theme
list()

$coordinates
<ggproto object: Class CoordCartesian, Coord, gg>
    aspect: function
    backtransform_range: function
    clip: on
    default: TRUE
    distance: function
    draw_panel: function
    expand: TRUE
    is_free: function
    is_linear: function
    labels: function
    limits: list
    modify_scales: function
    range: function
    render_axis_h: function
    render_axis_v: function
    render_bg: function
    render_fg: function
    reverse: none
    setup_data: function
    setup_layout: function
    setup_panel_guides: function
    setup_panel_params: function
    setup_params: function
    train_panel_guides: function
    transform: function
    super:  <ggproto object: Class CoordCartesian, Coord, gg>

$facet
<ggproto object: Class FacetWrap, Facet, gg>
    attach_axes: function
    attach_strips: function
    compute_layout: function
    draw_back: function
    draw_front: function
    draw_labels: function
    draw_panel_content: function
    draw_panels: function
    finish_data: function
    format_strip_labels: function
    init_gtable: function
    init_scales: function
    map_data: function
    params: list
    set_panel_size: function
    setup_data: function
    setup_panel_params: function
    setup_params: function
    shrink: TRUE
    train_scales: function
    vars: function
    super:  <ggproto object: Class FacetWrap, Facet, gg>

$plot_env
<environment: 0x7fc5d3ddc570>

$layout
<ggproto object: Class Layout, gg>
    coord: NULL
    coord_params: list
    facet: NULL
    facet_params: list
    finish_data: function
    get_scales: function
    layout: NULL
    map_position: function
    panel_params: NULL
    panel_scales_x: NULL
    panel_scales_y: NULL
    render: function
    render_labels: function
    reset_scales: function
    resolve_label: function
    setup: function
    setup_panel_guides: function
    setup_panel_params: function
    train_position: function
    super:  <ggproto object: Class Layout, gg>

$labels
$labels$x
[1] "Value of parameter"

$labels$title
[1] "Posterior plots with 95% credible intervals"

$labels$y
[1] "density"

$labels$xintercept
[1] "quants"

$labels$label
[1] "paste(\"Posterior mean: \", post.mean, \"\\nPosterior sd  : \", post.sd, ..."

$labels$vjust
[1] "vjust"

$labels$hjust
[1] "hjust"

$labels$fill
[1] "fill"
attr(,"fallback")
[1] TRUE

$labels$weight
[1] "weight"
attr(,"fallback")
[1] TRUE


attr(,"class")
[1] "gg"     "ggplot"

A plot of the \(logit(P)\) (the logit of the estimated probability of capture) is shown in ?@fig-btspas-diag1-logitP-nondiag

NULL

A summary of the posterior for each parameter is also available. The estimates of total abundance can be extracted and summarized in a similar fashion as in the other models:

BTSPAS.nondiag.est1 <-  Petersen::LP_BTSPAS_est (BTSPAS.nondiag.fit1)
BTSPAS.nondiag.est1$summary

  N_hat_f N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 337613.2 33764.97             0.95         Posterior
  N_hat_LCL N_hat_UCL p_model name_model cond.ll n.parms   nobs
1  285518.2  415926.8      ~1      p: ~1      NA      NA 150429

The estimated total abundance from BTSPAS is 337,613 (SD 33,765 ) fish.

9 Incomplete or partial stratification

9.1 Introduction

Capture heterogeneity is known to cause bias in estimates of abundance in capture–recapture studies. This heterogeneity is often related to observable fixed characteristics of the animals such as sex. If this information can be observed for every handled animal at both sample occasions, then it is straight- forward to stratify (e.g., by sex) and obtain stratum-specific estimates.

However, it may be difficult to measure the covariate for every fish. For example, it may be difficult to sex all captured fish because morphological differences are slight or because of logistic constraints. In these cases, a subsample of the captured fish at each sample occasion is selected, and additional and often more costly measurements are made, such as sex determination through sacrificing the fish.

The resulting data now consist of two types of marked animals: animals whose value of the stratification variable is unknown, and subsamples at each occasion where the value of the stratification variables are determined.

Premarathna, Schwarz, and Jones (2018) developed methodology for these types of studies. Furthermore, given the relative costs of sampling for a simple capture and for processing the subsample, optimal allocation of effort for a given cost can be determined. They also developed methods to account for additional information (e.g., prior information about the sex ratio) and for supplemental continuous covariates such as length.

These methods were applied to a problem of estimating the size of the walleye population in Mille Lacs Lake, MN and the data are reanalyzed here.

Note that this Petersen package functions for this methods are simple wrappers around the published code from the paper. Furthermore, wrapper are available only for model without continuous covariates in the Petersen package. Contact the above authors for code for more complex cases.

The complete statistical theory is presented in Premarathna, Schwarz, and Jones (2018).

9.2 Sampling protocol

At the first sample occasion, a random sample of size \(n_1\) is captured. Then, a subsample of size \(n^∗_1\) is selected from \(n_1\), and the stratum is determined for all animals in the subsample. All captured animals are tagged, usually with a unique tag number, and released back to the population.

Again, some time later, another sample of animals of size \(n_2\) is captured randomly from the population. The animals captured at the second sample occasion contain animals captured and marked at the first occasion (some of them might be stratified and some might not be stratified), as well as animals not captured at the first occasion. One of the requirements here is that some of the stratified subsamples at the first occasion are recaptured. Also, animals must not be sacrificed to determine stratification membership at the first sample occasion. Again, a subsample of size \(n^∗_2\) is selected from the captured sample at the second sample occasion including only animals not marked at the first occasion. A pictorial view of the sampling protocol is given in Figure 24

Figure 24: Schematic of sampling protocol for incomplete stratification studies

Note that only unmarked fish are examined in the subsample at both sampling occasions

9.3 Capture histories and parameters

The parameters for this type of study are

• \(N\) - population abundance
• \(p_1\), \(p_2\) - capture probabilities at the two sampling occasions
• \(\lambda_1\),…\(\lambda_k\) the proportion of the population in the \(k\) categories. Must add to 1.
• \(\theta_1\), \(\theta_2\) - the subsampling proportions at the two occasions.

There are \(3k+4\) possible capture histories. Each history is a two-digit string with entries of 0 (not captured), U (stratification unknown), or a code for the stratification variable (e.g. M, F, etc.).

The possible histories and corresponding probabilities are:

History	Probability
MM	\(\lambda_{M}\ p_{1M} \ \theta_1 \ p_{2M}\)
M0	\(\lambda_{M} \ p_{1M} \ \theta_1 \ (1-p_{2M})\)
0M	\(\lambda_{M} \ (1-p_{1M})\ p_{2M}\ \theta_{2}\)
\(\ldots\)	Similarly for other categories
UU	\(\lambda_{M} \ p_{1M}\ (1-\theta_1)\ p_{2M} + \lambda_{F} \ p_{1F} \ (1-\theta_1) \ p_{2F}\)
U0	\(\lambda_{M} \ p_{1M}\ (1-\theta_1)\ (1-p_{2M}) + \lambda_{F} \ p_{1F} \ (1-\theta_1) \ (1-p_{2F})\)
0U	\(\lambda_{M} \ (1-p_{1M}) \ p_{2M} \ (1-\theta_2) + \lambda_{F} \ (1-p_{1F}) \ p_{2F} \ (1-\theta_2)\)
00	Everything else (not observable)

Note that histories such as UF are not allowed as only unmarked fish in the second sample are sub-sampled. Also histories such as MF would indicates an error.

Standard numerical likelihood methods are applied in the usual way to estimate the parameters and obtain measures of uncertainty. After estimates are obtained, stratum specific abundance estimates are obtained as \(\widehat{N}_k = \widehat{N} \times \widehat{\lambda}_k\) and measures of uncertainty obtained using the delta method.

9.4 Example - Walleye in Milles Lac, MN.

This is the data analyzed in Premarathna, Schwarz, and Jones (2018). There are some slight differences from the data in the published paper because fish that did not have length also measured are removed.

Walleye are captured on the spawning grounds. Almost all the fish can be sexed in the first occasion All the captured fish are tagged and released and the recapture occurred 3 weeks later using gill-nets From a sample of fish captured at second occasion that are not tagged, a random sample is selected and sexed.

Here is the summary data:

data(data_wae_is_short)  # does not include length covariate
data_wae_is_short

  cap_hist freq
1       0F  237
2       0M   41
3       0U 3058
4       F0 1555
5       FF   32
6       M0 5071
7       MM   40
8       U0   42
9       UU    1

We start by fitting a model that allows for different probabilities of capture by sampling occasion and sex. Notice that we specify the “hidden” stratification variable using ..cat in the model for \(p\). It is possible to specify models for \(\lambda\) (category proportions) and \(\theta\) (subsampling fractions) but these are seldom useful (but see an example in the published paper).

Premarathna, Schwarz, and Jones (2018) provided a function to print the full results in a nice format

LP_IS_print(wae.res1)

Model information: 

Model Name:  p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0 
Neg Log-Likelihood:  -71018.68 
Number of Parameters:  8 
AICc value: -142021.4 

 

Raw data: 

$History
[1] "0F" "0M" "0U" "F0" "FF" "M0" "MM" "U0" "UU"

$counts
[1]  237   41 3058 1555   32 5071   40   42    1

$category
[1] "F" "M"


 

 
Initial values used for optimization routine: 

Initial capture probabilities: 
      time1      time2
F 0.0214139 0.01082925
M 0.0214139 0.01082925

Initial category proportions: 
lambda_ F lambda_ M 
      0.5       0.5 

Initial sub-sample  proportions: 
   theta_1    theta_2 
0.99362112 0.08333333 

Initial population size for optimization(simple Lincoln Petersen estimator is used) : 314,795 

 
Design matrix  and OFFSET vector for capture probabilities: 
    Beta...catF...time1 Beta...catM...time1 Beta...catF...time2
p1F                   1                   0                   0
p1M                   0                   1                   0
p2F                   0                   0                   1
p2M                   0                   0                   0
    Beta...catM...time2 OFFSET.vector
p1F                   0             0
p1M                   0             0
p2F                   0             0
p2M                   1             0

Design matrix and OFFSET vector for sub-sample proportions (theta): 
        Beta...time1 Beta...time2 OFFSET.vector
theta_1            1            0             0
theta_2            0            1             0

Design matrix and OFFSET vector for category proportions (lambda):
         ..catF OFFSET.vector
lambda_F      1             0


Find MLEs: 

MLEs for capture probabilities: 
       time1       time2
F 0.01146535 0.020643986
M 0.07658161 0.007930921

MLEs for category proportions: 
lambda_ F lambda_ M 
 0.674743  0.325257 

MLEs for sub-sample  proportions: 
  theta_1   theta_2 
0.9936211 0.0833333 

MLE for Population size :  206,507
MLE for Population size of category F : 139,339 
MLE for Population size of category M : 67,168
 
SE's of the MLEs 

SE's of the MLEs of capture probabilities: 
        time1       time2
F 0.002009477 0.003569036
M 0.015121929 0.001240106

SE's of the MLEs of the category proportions: 
 lambda_ F  lambda_ M 
0.06017088 0.06017088 

SE's of the MLEs of the sub-sample  proportions: 
    theta_1     theta_2 
0.000969662 0.004785221 

SE of the MLE of the population size:  26,475
SE for Population size of category F : 24,172 
SE for Population size of category M : 13,234

Observed and Expected counts for capture histories for the model  p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0 
  History Observed.Counts Expected.counts     Residual Standardized.Residuals
1      U0              42      42.5318920 -0.531891968           -0.081566309
2      UU               1       0.4706046  0.529395437            0.771707324
3      F0            1555    1554.6107699  0.389230058            0.009909151
4      M0            5071    5070.4718173  0.528182724            0.007510317
5      FF              32      32.7698643 -0.769864335           -0.134496611
6      MM              40      40.5349888 -0.534988778           -0.084037335
7      0F             237     236.9610300  0.038969959            0.002533033
8      0M              41      40.9922556  0.007744408            0.001209708
9      0U            3058    3057.4874952  0.512504751            0.009338016

Standardized residual plot for the model p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0 

Estimates of abundance are found in the usual fashion, both for the entire population:

wae.est1 <- LP_IS_est(wae.res1, N_hat=~1, conf_level=0.95)
wae.est1$summary

  N_hat_f N_hat_conf_level N_hat_conf_method N_hat_rn    N_hat N_hat_SE
1      ~1             0.95              logN      All 206506.9 26474.59
  N_hat_LCL N_hat_UCL            p_model  theta_model lambda_mode
1  160623.4  265497.4 ~-1 + ..cat:..time ~-1 + ..time ~-1 + ..cat
                                                                                 name_model
1 p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0
   cond.ll n.parms  nobs  method
1 71018.68       8 10077 full ll

and for the individual strata:

wae.est1a <- LP_IS_est(wae.res1, N_hat=~-1+..cat, conf_level=0.95)
wae.est1a$summary

        N_hat_f N_hat_conf_level N_hat_conf_method N_hat_rn    N_hat N_hat_SE
1   ~-1 + ..cat             0.95              logN        F 139339.1 24172.49
1.1 ~-1 + ..cat             0.95              logN        M  67167.8 13233.86
    N_hat_LCL N_hat_UCL            p_model  theta_model lambda_mode
1    99176.04 195766.78 ~-1 + ..cat:..time ~-1 + ..time ~-1 + ..cat
1.1  45651.13  98825.89 ~-1 + ..cat:..time ~-1 + ..time ~-1 + ..cat
                                                                                   name_model
1   p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0
1.1 p: ~-1 + ..cat:..time;  theta: ~-1 + ..time;  lambda: ~-1 + ..cat;  offsets 0,0,0,0,0,0,0
     cond.ll n.parms  nobs  method
1   71018.68       8 10077 full ll
1.1 71018.68       8 10077 full ll

As with the other functions in this package, we can fit other models such as a pooled-Petersen model:

and compare models using AICc (Table 25)

aic.table <- LP_AICc(wae.res1, wae.res2)

Table 25: Model comparison in incomplete stratification fits

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~-1 + ..cat:..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	71,018.68	8	10,077	-142,021.35	0.00	1.00
p: ~-1 + ..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	70,786.44	6	10,077	-141,560.86	460.49	0.00

and it is clear that the pooled-Petersen has no support.

Model-averaging is unnecessary given the negligible support for the second fitted model, but again can be done in the usual fashion for the entire population abundance (Table 26):

ma.table <- LP_modavg(wae.res1, wae.res2)

Table 26: Model averaging in incomplete stratification fits

Modnames	AICcWt	Estimate	SE
p: ~-1 + ..cat:..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	1.00	206,507	26,475
p: ~-1 + ..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	0.00	314,672	36,232
Model averaged		206,507	26,475

and for the individual categories (Table 27).

ma.table <- LP_modavg(wae.res1, wae.res2, N_hat=~-1+..cat)

Table 27: Model averaging in incomplete stratification fits for individual sex srata

Modnames	AICcWt	Estimate	SE
p: ~-1 + ..cat:..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	1.00	139,339	24,172
p: ~-1 + ..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	0.00	82,275	9,617
Model averaged		139,339	24,172
p: ~-1 + ..cat:..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	1.00	67,168	13,234
p: ~-1 + ..time; theta: ~-1 + ..time; lambda: ~-1 + ..cat; offsets 0,0,0,0,0,0,0	0.00	232,398	26,810
Model averaged		67,168	13,234

9.5 Optimal allocation

In an incomplete-stratified two-sample capture–recapture study, there is a cost to capturing an animal at each sample occasion, a cost to identify the category of the captured animal in the subsamples, and also a fixed cost regardless of the sample size. If there is a fixed amount of funds (\(C_0\)) to be used in the study, then the objective is to find the optimal number of animals to capture at both sample occasions and the optimal sizes of the subsamples to be categorized so that the variance of the estimated population abundance (\(Var(\widehat{N})\)) is minimized.

The total cost (\(C\)) of the study can be considered as a linear function of sample sizes and is given by \[C = c_f+ n_1 c_1 + n^∗_1 c^∗_1 + n_2c_2 +n^∗_2c^∗_2 ≤ C_0\] where

• \(n^*_1 \le n_1\)
• \(n^*_2 \le n_2 - E(n_{UU}+\sum_C{n_{cc}})\)

Numerical optimization methods can be used to find the optimal allocation of \(n_1\), \(n_2\), \(n^∗_1\), and \(n^*_2\) with respect to the linear constraint such that Var(\(\widehat{N}\)) is minimized.

The following costs were considered for the analysis of optimal allocation:

• \(c_f = 0\),
• \(c_1 = 4\),
• \(c^∗_1 = 0.4\),
• \(c_2 = 6\),
• \(c^∗_2 = 0.4\), and
• \(C_0 = 90,000\).

These costs are the time in minutes for the operation on a fish. A value of \(C_0 = 90,000\) minutes (i.e., 1,500 h) available for this study.

Optimal allocation suitable guesstimates for the parameters using previous studies or the researcher’s experience. We used the following guesstimates for the parameters:

• \(N=209,000\),
• \(\lambda_M =0.33\),
• \(r_1 =6.5\), and
• \(r_2 =0.4\)

where \(r_1\) is the guesstimate of the ratio of \(p_{1M}∕p_{1F}\), and \(r_2\) is the guesstimate of the ratio of \(p_{2M}∕p_{2F}\).

It is difficult to give guesstimates for capture probabilities for each category at each sample occasion. However, in practice, we can give a ratio of the capture probabilities at each sample occasion. For example, if detection probability of males is half that of females at the first sample occasion, then the ratio \(r_1\) is 0.5. The guesstimates of the ratios \(r_1\) and \(r_2\) given above are calculated using the estimates for capture probabilities see previously.

Optimal allocations of sample sizes and subsample sizes produced by numerical methods for the given costs are \(n_1 = 8,929\), \(n^∗_1 =8,908\), \(n_2 =8,359\), and \(n^∗_2= 1,412\). At the optimal allocation, \(SE(\widehat{N})=13,657\) which is 50% lower than the SE obtained using the current allocation.

Different solutions are available for \(n_1\), \(n_2\), \(n^∗_1\), and \(n^∗_2\) at optimal allocation. Conditional contour plots were used to see these different solutions. A conditional contour plot in this situation is a contour plot for the standard error of \(\widehat{N}\) where two values of \(n_1\), \(n_2\), \(n^∗_1\), or \(n^∗_2\) are fixed at the optimal values. A conditional contour plot for standard error of \(\widehat{N}\)̂ when \(n^∗_1\) and \(n^∗_2\) are fixed at the optimal values and when \(n_1\) and \(n_2\) are fixed at the optimal values are given in Figure 25. These contour plots show that many solutions are possible for optimal allocation.

Figure 25: Conditional contour plot for SE of estimated abundance. o=optimal allocation; x=current allocation (outside plot boundaries)

Full \(R\) code is available from the paper authors.

9.6 Additional details

Premarathna, Schwarz, and Jones (2018) also provide details on

• determining the sample size for detecting differential catchabilities (power analysis);
• determining the approximate standard error to be expected under proposed sample sizes (sample size analysis),
• performing goodness-of-fit test for the chosen model,
• using individual covariates (such a length) and an conditional likelihood approach, and
• using a Bayesian analysis to incorporate outside information (such as the sex ratio).

Please consult their paper for more details.

We need to consider the issue of nonidentifiability in model fitting. All the parameters can be estimated in two-sample capture–recapture studies similar to Lincoln–Peterson (Williams et al., 2002) model fitting. However, we would not be able to estimate some of the parameters, for example, if no females were observed. There is no nonidentifiable issue on model fitting with walleye data because both males and females were observed. However, there can be a nonidentifiability issue with the population abundance parameter when modeling with individual heterogeneity, as described by Link (2003).

10 Double tagging studies to account for tag loss or non-reporting

10.1 Introduction

Tag loss is one of the few violations of assumptions that can be most easily accounted for by a modification of the sampling protocol.

Tag loss leads to a positive bias in estimates of population size because fewer tagged fish are “apparently” recovered, i.e., fish that lost tags look as if they are untagged. In order to estimate and to adjust for tag loss, some (or all) of the fish released must be double tagged.

The analysis of double tagging studies was described by Gulland (1963), Beverton and Holt (1957), Seber and Felton (1981) and Hyun et al (2012). Seber (1982) and Weatherall (1982) have a nice summary of dealing with tag loss and ad hoc adjustments to estimates of abundance. It is generally assumed that a fish that loses it tags is indistinguishable from fish that were not tagged originally.

This double tagging generally takes two forms:

• two identical or two different tags are applied. Either or both could be lost, and the retention probability could be different for each tag.
• a permanent mark (such a fin clip or a permanent dye injection or a ‘genetic tag’) is a applied. Fish that are recovered with the permanent mark but without a tag are known to have lost a tag, but the individual cannot be determined.

When double tagging fish, tag 1 of the double tag should be the same type and placed in the same location on the fish as singly tagged fish. If tags have tag numbers inscribed on the fish, you can identify from a recovery of a fish with a single tag, if it was singly or doubly tagged. Hyun et al (2012) used a different colored tags for the double tagged fish so that a fish with a single tag captured at the second event can be classified as either initially singly or doubly tagged.

The second tag can be the same tag type placed in a different location, a different tag (e.g. a PIT tag), or a permanent batch mark.

Models with a permanent double-tag are simplification of the models with tags than can be lost where the tag retention probability is set to 1. Some care is needed if a batch mark is used because you may also need to know the stratum of release. If the stratum of release is based of fish characteristics, then these can usually be measured at the second event, but if the stratum of release is based on geographic or temporal strata, then the batch mark may not be sufficient to identify the stratum of release.

Tag loss has been traditionally divided into two types (Beverton and Holt, 1957):

• Type I losses are losses that occur immediately after tagging, e.g. immediate tag shedding. This type of tag loss reduces the effective number of tags initially put out.
• Type II losses are those that happen steadily and gradually over an extended period of time following release of the tagged fish.

In a simple Petersen study, the two types of tag loss cannot be distinguished and the total tag loss between the two sampling occasions is estimated. If the exact time at liberty of tags is known, then Weatherall (1982) reviews how to estimate the two types of losses. If multiple sampling occasions take place (i.e., a Jolly-Seber study), Arnason and Mills (1981), and McDonald, Amstrup and Manly (2003) examine the biases that can be introduced, and Cowen and Schwarz (2006) develops methodology to correct for tag loss.

The Petersen package can deal with three types of double tagging studies:

• Two indistinguishable double tags are used so that if a fish is recovered with only one of the double tags, you do not know which tag was lost. This type of studies often occurs when fish are not directly handled so tag numbers inscribed on the tag cannot be read. Similarly, the tags must placed close together so that it is difficult to distinguish. If for example, double tags were placed on the left and right of a fish, or front and back of a fish, then this may an example of the second case.
• Two distinguishable double tags are used so that if a fish is recovered with only one of the double tags, you do know which of the double tags were lost. For example, every tag could have a unique tag number; or applied to the left/right side of a fish
• A permanent batch mark (e.g., fin clip) is applied that is assumed cannot be lost.

10.2 Capture histories

The apparent capture history of a fish is again denoted using \(\omega\). It is important to remember that the apparent capture-history may not reflect the underlying actual capture history. For example, a tagged fish could lose all its tags before being recaptured at the second event. This capture looks like the capture of an untagged fish, but actually is a double counting of a fish with a different apparent capture history.

The apparent capture history is expanded to also reflect the observed tagging status. A two-digit code is code for each sample event, with the first digit (0 or 1) indicating if tag 1 is present on the fish and the second value (0 or 1) indicating if tag 2 is present on the fish.

Each component of the tag-history vector is generalized to represent the status of each tag at each capture occasion Cowen and Schwarz (2006). For example, here are the possible history vectors when double tagging is performed:

For fish that received a single tag, possible histories are:

• (10 10): Fish single tagged with tag type \(A\) at occasion 1 and recaptured at sampling occasion 2.
• (10 00): Fish single tagged with tag type \(A\) at occasion 1 that are “not seen” at occasion 2. Note that this consists of fish which have retained their tag and were not captured at occasion 2 and fish that lost their tag and were recaptured at occasion 2. The latter are not recognizable has having being previously tagged.

For fish that received two indistinguishable double tags, possible histories are:

• (11 00): Fish double tagged but not recaptured with at least one of its tags present.
• (11 1X): One of the double tags was present, but it is not known which
• (11 11): Fish double tagged and recovered with both tags present.

For fish that received two distinguishable double tags, the history (11,1x) is divided into

• (11 10): Fish double tagged with tag types \(B\) at occasion 1, and recaptured only with the double tag in location 1
• (11 01): Similar to above, but now recaptured with double tag in location 2 only present

For fish that received a permanent second tag, possible histories are:

• (1P 00): Fish double tagged, but not recaptured with at least one of its tags present.
• (1P 0P): Fish double tagged, but one tag (non permanent) has been lose
• (1P 1P): Fish double tagged, and recovered with both tags present.

Finally, we have fish that apparently are recaptured for the first time at the second event.

• {00 10}: Fish that apparently are seen for the first time at the second sampling occasion. This includes fish captured for the first time at the second occasion, and also fish that were tagged at occasion 1, lost their tag(s) between the two sampling occasions, and were recaptured at occasion 2.

Now, in the case with two distinguishable tags \[n_1 = n_{1010} + n_{1000} + n_{1111} + n_{1101} + n_{1110} + n_{1100}\] \[n_2 = n_{1010} + n_{1101} + n_{1110} + n_{1111} + n_{0010}\]

but a complete count of recaptured fish cannot be made because some fish lost all their tags and were captured and so have an apparent history of (00 10).

Similar equations can be derived for the other two cases.

The number of each fish in the study with the observed capture history is the frequency count and is denoted as \(n_{\omega}\).

The additional parameters to be introduced into the model are the tag retention probabilities at the two locations (\(\rho_1\), and \(\rho_2\)) and the proportion of fish that are single tagged at the first sampling occasion, \(p_{ST}\) which is assumed to be known given the ratio of single and double tagging done at the first sampling occasion.

A complete likelihood analysis based on capture histories is quite complex because of fish that loose all their tags and are recaptured. Hyun et al (2012) developed a likelihood-approach based on the summary statistics, but this does not easily allow for stratification, or if the capture and/or tagloss retention probabilities depend on covariates. In Appendix C we present a conditional likelihood approach similar to the approach used in the conditional likelihood Petersen estimator.

In this conditional likelihood approach we condition on fish captured at the second event and ignore histories of fish tagged and never seen again. This is a sensible approach because we cannot estimate the capture probability at the second event because the apparent untagged animals captured at the second occasion are mixture of truly untagged fish and fish that were captured at the first sampling occasion and lost all their tags. Then a HT-type estimator is used to estimate the abundance using the number of fish captured at the first occasion. Consequently, you can only develop a model for the catchability at the first sampling event (e.g., depends on discrete or continuous covariates).

• try and double tag as many fish as possible, esp if tag retention is low
• one of the double tags should match the single tag other wise you have 3 tag types and not much can be done.
• hypothesis test for tagging of the form “no tag loss” are silly. If any tag loss is observed, then then hypothesis is obviously rejected. If tag loss is very low, then you can fit a simple Petersen study and live with the small bias.

10.3 Example - Kookanee on Metolius River - two tags that cannot be distinguished

This is the data analyzed in Hyun et al (2012).

In August and September 2007, the period just before the spawning run, adult kokanee were collected by beach seining in the upper arm of the lake near the confluence with the Metolius River. Fish were tagged with nonpermanent, plastic T-bar anchor tags and then were released back into the lake. Randomly selected fish received single tags of one color, while the other fish received two tags of a second color (i.e., the double tags were identical in color). In late September through October, spawning ground surveys were conducted by 2–3 people walking abreast in a downstream direction (or floating, in sections where the water depth and flow were too great to allow walking). Instead of being physically recaptured, the fish were resighted as they swam freely in the clear, relatively shallow water within the spawning areas of the river. The total number of fish observed with or without a tag (or tags) was recorded for each section, and information on the number and color of tags for each marked fish was also noted.

Notice that tags where only a single tag is applied to a fish are a different color than tags where double tags are applied so that a fish that is subsequently detected with a single tag can be identified if it originally had a one or two tags applied.

The data were extracted from Hyan et al (2012) and converted to capture-histories.

data(data_kokanee_tagloss)
data_kokanee_tagloss[,1:2]

  cap_hist  freq
1     1000  2589
2     1010   218
3     1111    35
4     111X    24
5     1100   432
6     0010 11167

Because fish were not handled, it is not possible to know which of the double tags were lost, (i.e., we cannot distinguish between histories 1110 and 1101 and these are pooled into capture history 111X). Only models with equal retention probabilities for the two tags can be fit.

kok.res1 <- LP_TL_fit(data_kokanee_tagloss, 
                      dt_type="notD", 
                      p_model=~1, 
                      rho_model=~1)
kok.res1$summary

  p_model rho_model      name_model   cond.ll n.parms  nobs     method
1      ~1        ~1 p: ~1;  rho: ~1 -1488.626       2 11420 TL cond ll

Estimates of abundance are:

kok.est1 <- LP_TL_est(kok.res1, N_hat=~1, conf_level=0.95)
kok.est1$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 103035.9 9099.521             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model       name_model   cond.ll n.parms
1  86659.33  122507.2      ~1        ~1 p1: ~1;  rho: ~1 -1488.626       2
   nobs        method
1 14465 LP_TS CondLik

The conditional likelihood estimate is 103 (SE 9) thousand fish compared to the estimates of 103 (SE 9) thousand fish obtained by Hyan et al (2012).

Estimates of the tag retention probability is

kok.est1$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho     rho.se
2       1 0.7266325 0.05117941
2.1     2 0.7266325 0.05117941

These estimates match those from Hyan et al (2012) of tag retention of \(1-.27=.73\) (SE .05).

In this example, we conditioned on being captured at time 2 so avoid double counting fish that were tagged, lost all tags, and then were recaptured.

It is estimated that 85 single tagged fish lost their tag and were recaptured at the second event, and 4 double tagged fish lost all their tags and were recaptured at the second event. If these are added to the 277 OBSERVED recaptures at the second event, it is estimated that the total number of recaptures (observed and unobserved) is 366 fish. The regular-Petersen estimator using the observed values of \(n_1=\) 3298 and \(n_2=\) 1.1444^{4} is \(\widehat{N}_{Petersen}=\) 103,036 fish which matches very closely (as it must) to the estimates from the model. Of course, the estimated SE from the Petersen estimator using the estimated number of recaptures will be too small because it does not account for the uncertainty in estimating the expected number of recaptures.

10.4 Example - Simulated data - two tags that can be distinguished

Simulated data are used for this example. Tried but could not find numerical example in literature

The observed capture history data is:

data(data_sim_tagloss_twoD)
data_sim_tagloss_twoD[,-(1:2)]

  cap_hist freq
1     0010  879
2     1000  225
3     1010   14
4     1100  666
5     1101   21
6     1110    7
7     1111   37

Notice that we now have histories (11,01) and (11,10) compared to the previous example where these were “combined” into history (11,1X).

We can not fit models with and without equal tag retention probabilities and compare them in the usual fashion.

10.4.1 Model with equal tag retention probabilies

First, the model with equal retention probabilities across the two tag locations:

twoD.res1 <- LP_TL_fit(data_sim_tagloss_twoD, 
                       dt_type="twoD", 
                       p_model=~1, 
                       rho_model=~1)
twoD.est1 <- LP_TL_est(twoD.res1, N_hat=~1, conf_level=0.95)
twoD.est1$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 10267.11 1214.854             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model       name_model   cond.ll n.parms nobs
1  8141.981  12946.92      ~1        ~1 p1: ~1;  rho: ~1 -373.7071       2 1849
         method
1 LP_TS CondLik

Estimates of the tag retention probability is

twoD.est1$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho     rho.se
1       1 0.7213776 0.04992898
1.1     2 0.7213776 0.04992898

10.4.2 Model with unequal tag retention probabilies

Second, a model with unequal retention probabilities across the two tag locations is fit

twoD.res2 <- LP_TL_fit(data_sim_tagloss_twoD, 
                       dt_type="twoD", 
                       p_model=~1, 
                       rho_model=~-1+..tag)
twoD.est2 <- LP_TL_est(twoD.res2, N_hat=~1, conf_level=0.95)
twoD.est2$summary

  N_hat_f    N_hat_rn   N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 10196.2 1194.993             0.95              logN
  N_hat_LCL N_hat_UCL p_model   rho_model                name_model   cond.ll
1  8103.591  12829.18      ~1 ~-1 + ..tag p1: ~1;  rho: ~-1 + ..tag -369.8691
  n.parms nobs        method
1       3 1849 LP_TS CondLik

Note the use of the ..tag variable in the formula for rho indicating that tag location is a factor that influences the retention probability.

Estimates of the different tag retention probabilities are

twoD.est2$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho     rho.se
1       1 0.6363982 0.06085315
1.1     2 0.8409085 0.05514061

10.4.3 Comparing models and model averaged estimates of abundance

We compare the two models in the usual way (Table 28) and obtain the model averaged estimates of abundance (Table 29).

sim.twoD.aictab <- LP_AICc(
  twoD.res1,
  twoD.res2
)

Table 28: Comparison of models fit to the simulated tag loss dataset with distinguishable table

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~1; rho: ~-1 + ..tag	-369.8691	3	958	745.76	0.00	0.94
p: ~1; rho: ~1	-373.7071	2	958	751.43	5.66	0.06

Most weight is given to the model with different tag retention probabilities.

# extract the estimates of the abundance 
sim.twoD.ma.N_hat<- LP_modavg(
  twoD.res1,
  twoD.res2,  N_hat=~1)  

Table 29: Model averaged estimates of abundance for tag loss model with distinguishable tags

Modnames	AICcWt	Estimate	SE
p1: ~1; rho: ~-1 + ..tag	0.94	10,196	1,195
p1: ~1; rho: ~1	0.06	10,267	1,215
Model averaged		10,200	1,196

10.5 Example - Simulated data - a permanent second tag used

Simulated data are used for this example. Tried but could not find numerical example in literature

The observed capture history data is:

data(data_sim_tagloss_t2perm)
data_sim_tagloss_t2perm[,-(1:2)]

  cap_hist freq
1     0010  923
2     1000  254
3     1010   17
4     1P00  668
5     1P0P   26
6     1P1P   59

We can now fit models but the model for the tag retention probability cannot depend on the tag number since the permanent tag has retention probability of 1.

t2perm.res1 <- LP_TL_fit(data_sim_tagloss_t2perm, 
                         dt_type="t2perm", 
                         p_model=~1, 
                         rho_model=~1)
t2perm.est1 <- LP_TL_est(t2perm.res1, N_hat=~1, conf_level=0.95)
t2perm.est1$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 9425.522  937.881             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model       name_model   cond.ll n.parms nobs
1  7755.452  11455.23      ~1        ~1 p1: ~1;  rho: ~1 -430.7471       2 1947
         method
1 LP_TS CondLik

Estimates of the probability of capture at the first event, and the tag retention probabilities are

t2perm.est1$detail$data[1, c('p1',"p1.se")]

         p1      p1.se
1 0.1086412 0.01032415

t2perm.est1$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho     rho.se
1       1 0.6824881 0.04923361
1.1     2 1.0000000 0.00000000

10.6 Example - Northern Pike - tag loss

In Section 6.2, we analyzed the Northern Pike data assuming that tag loss was negligible. In this example, we will estimate the tag retention probability and show that the conditional likelihood tagloss models can include more complex effects.

Refer to Section 6.2 for information on the sampling design.

We load the data in the usual fashion:

data(data_NorthernPike_tagloss)
data_NorthernPike_tagloss[1:5,]

  cap_hist Sex Length freq
1     0010   m  23.20    1
2     0010   f  28.89    1
3     0010   m  25.20    1
4     0010   m  22.20    1
5     0010   m  25.00    1

Fish with histories 1110 and 1101 are pooled into capture history 111X and only models with equal retention probabilities for the two tags can be fit.

We start with a basic model, where capture probabilities and tag-retention probabilities don’t vary over the fish. The estimate of abundance is:

nop.TL.res1 <- LP_TL_fit(data_NorthernPike_tagloss, 
                         dt_type="notD", 
                         p_model=~1, 
                         rho_model=~1)
nop.TL.est1 <- LP_TL_est(nop.TL.res1, N_hat=~1, conf_level=0.95)
nop.TL.est1$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 49516.75 3711.682             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model       name_model   cond.ll n.parms nobs
1  42751.13  57353.06      ~1        ~1 p1: ~1;  rho: ~1 -482.3847       2 7805
         method
1 LP_TS CondLik

Estimates of the tag retention probability is

nop.TL.est1$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho      rho.se
1       1 0.9805195 0.007951377
1.1     2 0.9805195 0.007951377

So estimated tag retention was over 98%, and that is why tag loss was previously ignored.

We now fit a model stratified by sex and the estimate of overall abundance is now:

nop.TL.res2 <- LP_TL_fit(data_NorthernPike_tagloss, 
                         dt_type="notD", 
                         p_model=~Sex, 
                         rho_model=~1)
nop.TL.est2 <- LP_TL_est(nop.TL.res2, N_hat=~1, conf_level=0.95)
nop.TL.est2$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 49363.51 3699.588             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model         name_model   cond.ll n.parms
1  42619.86  57174.19    ~Sex        ~1 p1: ~Sex;  rho: ~1 -482.0723       3
  nobs        method
1 7805 LP_TS CondLik

Finally, we allow the tag retention probabilities to also vary by sex and the estimated overall abundance is:

nop.TL.res3 <- LP_TL_fit(data_NorthernPike_tagloss, 
                         dt_type="notD", 
                         p_model=~Sex, 
                         rho_model=~Sex)
nop.TL.est3 <- LP_TL_est(nop.TL.res3, N_hat=~1, conf_level=0.95)
nop.TL.est3$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 49363.12 3699.555             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model           name_model  cond.ll n.parms
1  42619.53  57173.73    ~Sex      ~Sex p1: ~Sex;  rho: ~Sex -482.016       4
  nobs        method
1 7805 LP_TS CondLik

Estimates of the tag retention probability for the two sexes are

firstm <- match("m", nop.TL.est3$detail$data.expand$Sex) 
firstf <- match("f", nop.TL.est3$detail$data.expand$Sex)
nop.TL.est3$detail$data.expand[sort(c(firstm, firstm+1, firstf, firstf+1)),c("Sex","..tag","rho","rho.se")]

    Sex ..tag       rho      rho.se
1     m     1 0.9774436 0.013019620
1.1   m     2 0.9774436 0.013019620
2     f     1 0.9828572 0.009895976
2.1   f     2 0.9828572 0.009895976

The tag retention probabilities are very similar across the two sexes.

The usual model averaging can be done as shown in Table 30.

nop.TL.aictab <- Petersen::LP_AICc (
      nop.TL.res1,
      nop.TL.res2,
      nop.TL.res3
)

Table 30: Model comparison of models fit to Northern Pike double tagging data

Model specification	Conditional log-likelihood	# parms	# obs	AICc	Delta	AICcWt
p: ~1; rho: ~1	-482.3847	2	1,134	968.78	0.00	0.59
p: ~Sex; rho: ~1	-482.0723	3	1,134	970.17	1.39	0.30
p: ~Sex; rho: ~Sex	-482.0160	4	1,134	972.07	3.29	0.11

Most of the model weight is on the simplest model. The model averaged overall abundance estimate are (Table 31)

nop.TL.modavg <- Petersen::LP_modavg (
      nop.TL.res1,
      nop.TL.res2,
      nop.TL.res3
)

Table 31: Model average estimate of overall abundance from models fit to Northern Pike double tagging data

N_hat_f	N_hat_rn	Modnames	AICcWt	Estimate	SE
~1	(Intercept)	p1: ~1; rho: ~1	0.59	49,517	3,712
~1	(Intercept)	p1: ~Sex; rho: ~1	0.30	49,364	3,700
~1	(Intercept)	p1: ~Sex; rho: ~Sex	0.12	49,363	3,700
		Model averaged		49,454	3,707

10.7 Example - Simulated data - non-reporting of tags and reward tags

One assumption of the Petersen estimator is that all recovered tags are reported. However, some studies relies on returns from anglers or other citizens, and not all tags may be reported. To estimate the reporting probability, a second type of tag (the “reward” tag) is applied that often has a monetary incentive to return. The monetary incentive should be large enough that the reporting probability for these tags is 100%.

These types of studies can also be analyzed using the tagloss models. The reward tag is treated as a permanent tag.

Simulated data are used for this example. The observed capture history data is:

data(data_sim_reward)
data_sim_reward

  cap_hist freq
1     0010  930
2     0P00  207
3     0P0P   23
4     1000  712
5     1010   56

In this example, about 1500 non-reward tags were applied, and about 230 reward tags were applied. No fish has both types of tags. The reporting probability is estimated by the ratio of the tags returned

# estimate return probability and empirical SE
t1.applied  <- sum(data_sim_reward$freq[ data_sim_reward$cap_hist %in% c("1000","1010")])
t1.returned <- sum(data_sim_reward$freq[ data_sim_reward$cap_hist %in% c(       "1010")])
t2.applied  <- sum(data_sim_reward$freq[ data_sim_reward$cap_hist %in% c("0P0P","0P00")])
t2.returned <- sum(data_sim_reward$freq[ data_sim_reward$cap_hist %in% c("0P0P"       )])

rr    <- (t1.returned/t1.applied)  / (t2.returned/t2.applied)
rr.se <- sd( rbeta(10000, t1.returned, t1.applied-t1.returned)/ rbeta(10000, t2.returned, t2.applied-t2.returned))

and is estimated as 0.729 (SE 0.189).

We can now fit the permanent tag models as seen previously and obtain estimates of abundance:

reward.res1 <- LP_TL_fit(data_sim_reward, 
                         dt_type="t2perm", 
                         p_model=~1, 
                         rho_model=~1)
reward.est1 <- LP_TL_est(reward.res1, N_hat=~1, conf_level=0.95)
reward.est1$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 10090.01 2101.777             0.95              logN
  N_hat_LCL N_hat_UCL p_model rho_model       name_model   cond.ll n.parms nobs
1  6707.858  15177.46      ~1        ~1 p1: ~1;  rho: ~1 -324.7077       2 1928
         method
1 LP_TS CondLik

Estimates of the probability of capture at the first event, and the tag reporting probability are

reward.est1$detail$data[1, c('p1',"p1.se")]

          p1      p1.se
1 0.09890973 0.02038768

reward.est1$detail$data.expand[1:2,c("..tag","rho","rho.se")]

    ..tag       rho    rho.se
1       1 0.7291672 0.1805853
1.1     2 1.0000000 0.0000000

10.8 Planning tag loss studies

As a general rule, unless you have good information on tag retention probabilities, it is better to assume the worst and plan studies under the pessimistic assumption that tag loss will occur. The primary question is then how many fish should be double tagged?

This is a trade off because it is more costly to actually double tag fish, both when applying tags, but also in recording tag numbers, etc. There are additional tagging costs etc.

The easiest way to investigate the optimal sampling design is via simulation.

10.8.1 Example of deciding between proportion of double tagging

Suppose that you have $15,000 for tagging operations and that double tagging a fish costs 2x as much as single tagging a fish in supplies (more tags) and additional labor. We can simulate a tagging data set with various parameters, and then compare the precision (SE) as a function of the number of double tagged fish.

We start by making some assumptions about the

• abundance say, 10,000 fish.
• tag retention probabilities (say 0.80 for all fish)
• effort at the second sample, say 1000 fish can be examined.

We first generate the scenarios that we wish to compare

scenarios <- expand.grid(N=10000,
                         rho1=0.80,
                         rho2=0.80,
                         n2=1000,
                         n1.dt=seq(50, 500, 50),# number of double tags
                         n.sim=10)              # simulations per scenario
scenarios$n1.st <- 1500 - scenarios$n1.dt*3   # how many single tags can be applied
scenarios$n1 <- scenarios$n1.st + scenarios$n1.dt
scenarios$pST<- scenarios$n1.st/ scenarios$n1
scenarios$index <- 1:nrow(scenarios)
scenarios

       N rho1 rho2   n2 n1.dt n.sim n1.st   n1       pST index
1  10000  0.8  0.8 1000    50    10  1350 1400 0.9642857     1
2  10000  0.8  0.8 1000   100    10  1200 1300 0.9230769     2
3  10000  0.8  0.8 1000   150    10  1050 1200 0.8750000     3
4  10000  0.8  0.8 1000   200    10   900 1100 0.8181818     4
5  10000  0.8  0.8 1000   250    10   750 1000 0.7500000     5
6  10000  0.8  0.8 1000   300    10   600  900 0.6666667     6
7  10000  0.8  0.8 1000   350    10   450  800 0.5625000     7
8  10000  0.8  0.8 1000   400    10   300  700 0.4285714     8
9  10000  0.8  0.8 1000   450    10   150  600 0.2500000     9
10 10000  0.8  0.8 1000   500    10     0  500 0.0000000    10

As more fish are double tagged, the total number of tagged fish declines.

Now for each scenario, we simulate some data, analyze it using the notD tag loss model and plot the estimated precision vs number of double tags (Figure 26)

Figure 26: SE of N_hat vs. number of double tagged fish

The wiggliness of the line is due to the small number of simulations run for each scenario.

We see that the SE of \(\widehat{N}\) declines rapidly as you increase the number of double tagged fish to about 200, and then slowly increases because the gain in precision from estimating the tag retention probabilities better with more double tagged fish is offset by the loss in precision by tagging fewer fish.

11 Multiple marks

11.1 Introduction

In some cases, multiple marks can be applied to an animal, but not all marks can be read for an encountered animal. The most common example of this are capture-recapture studies using natural marks found on the left or right side of an animals and “captures” are photographs.

Unfortunately, use of photo-identification records in mark–recapture analyses is not entirely without its problems. For instance, when natural markings are bilaterally asymmetrical, matching photographs to individuals can be difficult when investigators are routinely able to photograph only one side of an animal. The typical approach in the literature when confronted with this situation is to conduct separate analyses with left-sided and right-sided photographs and compare the results. The same animal may be “captured” more than once in a sampling event, but the two records, from the right or left side of the animal, cannot be matched.

Bonner and Holmberg (2013) and McClintock et al. (2013) simultaneously presented the theory for these analyses and McClintock (2015) created an R package (multimark) for the analysis of these experiment.

Much of the following is taken from these papers as applied to a two-sample (Petersen-type) experiment.

11.2 Theory

Consider a “typical” Petersen capture–recapture experiment where sampling is conducted over 2 sampling occasions and the identity of each animal is known with certainty when it is observed based on applied tags or marks. As seen previously, there are four possible encounter histories, three of which are directly observable:

• 01 (encountered on the second occasion but not the first),
• 10 (encountered on the first occasion but not the second), and
• 11 (encountered on both occasions),
• 00 (not encountered on either sampling event)- not observable.

In contrast to the preceding scenario, now consider the situation in which individuals are encountered via photographs and marks are bilaterally asymmetrical.

We will consider two photo-sampling scenarios, where on each sampling event, a

• 0 represents a non-encounter,
• L represents an encounter on the left side only as L,
• R represents an encounter on right-sided only, and
• B represent an encounter on both sides. Depending on the design of camera surveys, B encounters may be recorded from a single image (where both sides of an individual are visible), a pair of synchronized images, or multiple images (assuming both sides of an individual are known.

One-sided analyses are common for bilateral encounter data when researchers are unable to match opposite sides, but are able to accurately match left sides to left sides and right sides to right sides. Under these conditions, attempts to combine left- and right-sided encounters into a single analysis are problematic because detected individuals can yield a number of possible recorded histories, depending on whether B encounters are observable. For example, when presented with the recorded histories L0 and 0R, we do not know whether these observations arose from the same animal seen on both occasions, or whether it was indeed two different animals each seen on one occasion. Similarly, when B detections are not observed, the recorded histories L0 and R0 could arise from a single animal or from two different animals.

This is a similar issue as seen when analyzing double tagging experiments with tag loss.

There are two different data types that can be collected:

• only the left or right side (LR data type) or
• the left, right, or both sides (LRB data type). Depending on the design of camera surveys, B encounters may be recorded from a single image (where both sides of an individual are visible), a pair of synchronized images, or multiple images (assuming both sides of an individual are known.

Now for any one particular animal, there are 16 possible (latent) encounter histories that give rise to the observable encounter histories (Table 32).

Table 32: Latent histories (column 2) and recorded histories (columns 6 and 9 ) from left- and right-side encounters only (LR) or from left-, right-, and both-side encounters (LRB) with 2 sampling events. Taken from McClintock et al (2013).

Column 4 in Table 32 indicates which underlying (latent) encounter histories are mapped to the observable encounter histories. For example, an animal not seen in the first sampling event, and then both sides seen the second sampling event (OB, forth row), gives rise to two observable (recorded) histories (0L and 0R) in experiment where both sides can never be matched to a single animal.

The parameters of this model are

• \(N\) abundance
• \(p_t\) encounter probability at sampling event \(t\)
• \(\delta^L\) probability that an individual which is encountered has its Left side photographed
• \(\delta^R\) probability that an individual which is encountered has its right side photographed
• \(\delta^B\) probability that an individual which is encountered has both sides photographed

These parameters can be used to compute the probability of each underlying (latent) encounter history (column 3 in Table 32). These probabilities must sum to 1 over all latent encounter histories. Now many of the latent encounter histories are mapped to multiple recorded histories. Their probabilities will no longer sum to 1 because many latent histories are double counted, and the counts from the reported histories (columns 6 and 9) no longer follow a multinomial distribution.

Because of this potential double counting, a likelihood analysis is very difficult and both Bonner and Holmberg (2013) and McClintock et al (2013) developed a Bayesian approach. Their papers should be consulted for details. McClintock (2015) developed an R package (multimark) that performs the Bayesian analysis using Monte Carlo Markov Chain (MCMC) methods which will be illustrated below.

By referring to Table 32, the expected counts for each of the capture histories under the LR protocol are:

• \(E[ n_{L0}] = N p_1 (1-\delta_R) (1-p_2 + p_2 \delta_R))\)
• \(E[ n_{LL}] = N p_1 (1-\delta_R) p_2 (1-\delta_R))\)
• \(E[ n_{0L}] = N (1-p_1+p_1\delta_R) p_2 (1-\delta_R))\)
• \(E[ n_{R0}] = N p_1 (1-\delta_L) (1-p_2 + p_2 \delta_L))\)
• \(E[ n_{RR}] = N p_1 (1-\delta_L) p_2 (1-\delta_L))\)
• \(E[ n_{0R}] = N (1-p_1+p_1\delta_L) p_2 (1-\delta_L))\)

Notice that \(\delta_L + \delta_R + \delta_B=1\).

Two Petersen estimators for abundance can be found by using the animals tagged on the Left or tagged on the Right exclusively, i.e.,

\[\widehat{N}_L = \frac{(n_{L0}+n_{LL}){(n_{0L}+n_{LL})}}{n_{LL}}\] \[\widehat{N}_R = \frac{(n_{R0}+n_{RR}){(n_{0R}+n_{RR})}}{n_{RR}}\] By substituting in the expected values, we see that these estimators are consistent for \(N\).

The two estimates could be averaged in some fashion, but computing the SE of the combined estimate is complex because of the correlation in the the estimates.

Unfortunately, it is not possible to estimate \(p_1\), \(p_2\), \(\delta_L\) or \(\delta_R\) individually. But we can estimate, for example

\[\widehat{p_1(1-\delta_R)} = \frac{n_{LL}}{(n_{LL}+n_{OL})}\] which is an estimate of “Left side” catchability at time 1.

We can also estimate \[\widehat{\frac{(1-\delta_R)}{(1-\delta_L)}} = \frac{(n_{LL}+n_{L0})}{(n_{RR}+n_{R0})} \]

which is ratio of the complement of the conditional probabilities of detection on Left or Right. But the individual values of \(\delta_L\) and \(\delta_R\) cannot be found.

The latter will have implications when using multimark. If for example, left or right conditional probabilities are equal (which seems sensible for many species and experimental setups), then any value for \(\delta_L=\delta_R\) between 0 and 0.5 gives a ratio of the complements equal to 1. Hence the \(\delta\) values are non-identifiable individually, and the program will converge to a value dictated by the prior distribution for \(\delta\). Similarly, the individual estimates of \(p_1\) and \(p_2\) will depend on the final value for the \(\delta\) values and may be nonsensical. But incredibly, estimates of \(N\) will be valid!

This will be illustrated using simulated data.

However, if allow for some animals to have both sides recorded, then not only are the estimates of abundance unbiased, but you also get unbiased estimates of catchability delta.

11.3 Illustration of using multimark with simulated data.

11.3.1 LR-type surveys

The multimark package has a simulation function to generate data. We will simulate a population of 5000 individuals with different capture probabilities at the two sample times, but equal \(\delta\) values. Notice that multimark uses \(\delta_1\) for \(\delta_L\), etc. In this simulation, no animals have both sides recorded.

logit <- function (p){ log(p/(1-p))}
expit <- function(theta){ 1/(1+exp(-theta))}

set.seed(234324)
N= 5000
p1= .5
p2 = .7 

test <- multimark::simdataClosed(
  N = N,
  noccas = 2,
  pbeta = logit(c(p1, p2)),
  tau = 0, # behavioural effects (i.e. c)
  sigma2_zp = 0,
  delta_1 = 0.4,
  delta_2 = 0.4,
  alpha = 0,
  data.type = "never",
  link = "logit"
)

The first few encounter histories under the LR protocol are

      [,1] [,2]
 [1,]    0    1
 [2,]    2    0
 [3,]    1    0
 [4,]    0    1
 [5,]    1    1
 [6,]    2    2
 [7,]    1    1
 [8,]    0    1
 [9,]    1    1
[10,]    0    1

We see that encounter matrix has capture histories with 1 representing a L capture and a 2 representing a R capture.

The reduced set of capture histories is:

test.ch <- as.data.frame(test$Enc.Mat)

test.ch.red <- plyr::ddply(test.ch,c("V1","V2"), plyr::summarize, 
                           cap_hist=paste0(V1[1],V2[1]),
                           freq=length(V1))
test.ch.red

  V1 V2 cap_hist freq
1  0  1       01 1475
2  0  2       02 1453
3  1  0       10  899
4  1  1       11  631
5  2  0       20  900
6  2  2       22  621

Notice that \(\delta_B = 1-\delta_1-\delta_2 = 0.2\) so for 20% of animals encountered at a sampling event, both the left and right are photographed, but you are unable to link those two photographs together (e.g. taken by different individuals). This implies that the total number of encounters is MORE than expected from the values of \(p_1\) and \(p_2\) by themselves:

Number of pictures of animals at t=1  3051  vs. expected number of  2500 

Number of pictures of animals at t=1  4180  vs. expected number of  3500 

There evidently some animals detected on both sides, but photos could not be matched. The simulation function also returns the actual encounter history matrix (not shown) that show that some animals are detected on both sides, but could not be matched across the two photographs.

We can get estimates of abundance based on the left or right photos individually in the same way as shown in this document.

For example, based on the left side photographs, the capture history matrix and frequency counts are:

  V1 V2 cap_hist freq
1  0  1       01 1475
3  1  0       10  899
4  1  1       11  631

    n1   n2  m2   p.recap        mf
1 1530 2106 631 0.4124183 0.2996201

This gives the estimate of abundance that is consistent, but estimates of catchability that are confounded with the \(\delta\) values.

left.fit <- Petersen::LP_fit(temp, p_model=~..time)
left.est <- Petersen::LP_est(left.fit)
left.est$summary[, c("N_hat","N_hat_SE","N_hat_conf_level","N_hat_LCL","N_hat_UCL")]

     N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1 5106.467 130.4088             0.95  4857.162  5368.568

left.est$detail$data.expand[1:2,c("..time","p")]

    ..time         p
1        1 0.2996200
1.1      2 0.4124183

Similarly, the capture histories and estimated abundance based on the right side photographs is found as:

temp <- test.ch.red[grepl('2', test.ch.red$cap_hist),]
temp$cap_hist <- gsub('2','1',temp$cap_hist)
temp

  V1 V2 cap_hist freq
2  0  2       01 1453
5  2  0       10  900
6  2  2       11  621

Petersen::LP_summary_stats(temp)

    n1   n2  m2  p.recap        mf
1 1521 2074 621 0.408284 0.2994214

right.fit <- Petersen::LP_fit(temp, p_model=~..time)
right.est <- Petersen::LP_est(right.fit)
right.est$summary[, c("N_hat","N_hat_SE","N_hat_conf_level","N_hat_LCL","N_hat_UCL")]

     N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1 5079.794 131.2457             0.95  4828.962  5343.656

right.est$detail$data.expand[1:2,c("..time","p")]

    ..time         p
2        1 0.2994217
2.1      2 0.4082842

A naive estimate that combines the two estimates is a simple average with a SE assuming (incorrectly) that these estimates are independent is:

# naive simple average
naive.N.avg <- (left.est$summary$N_hat + right.est$summary$N_hat)/2
cat("Naive estimate of abundance based on simple average ", round(naive.N.avg), "\n")

Naive estimate of abundance based on simple average  5093 

naive.N.avg.se <- sqrt((left.est$summary$N_hat_SE^2 + right.est$summary$N_hat_SE^2)/4)
cat("Naive SE of average is ", round(naive.N.avg.se), "\n")

Naive SE of average is  93 

We now use multimark to estimate abundance based on the left and right photographs combined. A model that allows for different capture probabilities at each sampling event is fit by setting up a design matrix. The multimark packages does a Bayesian analysis and so you must specify the number of MCMC iterations to perform etc. The fitting function fits multiple chains in parallel using multiple cores in your machine.

# use multimark to estimate abundance

# we set up my own design matrix, because mod.p=~time creates a design matrix 
# that has 3 columns rather than only 2
myCovs <- data.frame(mytime=c(0,1))
myCovs

  mytime
1      0
2      1

max.iter <- 100000
n.thin   <- 10

multimark.res <- file.path("Images","multimark-sim-LR.Rdata")
if( file.exists(multimark.res)){
    load(multimark.res)
}
if(!file.exists(multimark.res)){
 fit <- multimark::multimarkClosed(
  test$Enc.Mat,
  data.type = "never",
  covs = myCovs,
  mms = NULL,
  mod.p = ~mytime,
  mod.delta = ~type,
  parms = c("pbeta", "delta", "N"),
  nchains = 3,
  iter = max.iter,
  adapt = 10000,
  bin = 50,
  thin = n.thin,
  burnin = 2000)
 save(fit, file=multimark.res)
}

A summary of the MCMC output is:

summary(fit$mcmc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                        Mean        SD  Naive SE Time-series SE
pbeta[(Intercept)]    0.4077 5.817e-02 3.393e-04      0.0042681
pbeta[mytime]         1.1306 8.011e-02 4.672e-04      0.0051688
N                  5065.2233 1.031e+02 6.013e-01      8.6828559
delta_1               0.5010 6.404e-03 3.735e-05      0.0002343
delta_2               0.4953 6.438e-03 3.755e-05      0.0002414

2. Quantiles for each variable:

                        2.5%       25%       50%       75%     97.5%
pbeta[(Intercept)]    0.2948    0.3672    0.4081    0.4485    0.5191
pbeta[mytime]         0.9828    1.0742    1.1278    1.1846    1.2924
N                  4877.0000 4992.0000 5060.0000 5135.0000 5276.0000
delta_1               0.4883    0.4967    0.5011    0.5054    0.5134
delta_2               0.4824    0.4910    0.4953    0.4997    0.5077

# get the estimates of p and c
pc <- multimark::getprobsClosed(fit, link = "logit")
summary(pc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

       Mean      SD  Naive SE Time-series SE
p[1] 0.6005 0.01395 8.134e-05       0.001023
p[2] 0.8225 0.01756 1.024e-04       0.001430
c[2] 0.8225 0.01756 1.024e-04       0.001430

2. Quantiles for each variable:

       2.5%    25%    50%    75%  97.5%
p[1] 0.5732 0.5908 0.6006 0.6103 0.6269
p[2] 0.7881 0.8102 0.8227 0.8352 0.8552
c[2] 0.7881 0.8102 0.8227 0.8352 0.8552

We see that the estimate of abundance appears to be unbiased and has comparable uncertainty to our naive average of estimates of abundance based on the left or right photo individually.

But the estimates of catchability and \(\delta\) are severely biased. Notice that \(\delta_1\) and \(\delta_2\) move towards 0.5 based on the prior distribution for these parameters. Consequently because \(\delta_1\) is overestimated, \(1-\delta_1\) is under estimated, and so the estimated capture probabilities are overestimated..

We can look at the diagnostic plots of the MCMC chains:

plot(fit$mcmc, auto.layout=FALSE)

Everything appears to be ok, but only the estimates of abundance are sensible, and the estimates of \(p\) or \(\delta\) really have no meaning!

Given these issues of non-identifiability in the case of 2 sampling events, model averaging likely doesn’t make much sense.

It is not clear if these issues of identifiability also affect analyses with more than 2 sampling events, but I suspect that they do. Some careful analysis using multimark will be required. YMMV.

11.3.2 LRB-type surveys

We now generate simulated data but both sides of some animals can be observed (the B capture type). We will again simulate a population of 5000 individuals with different capture probabilities at the two sample times, but equal \(\delta\) values. About 0.20 of animals that are encountered have both sides read.

logit <- function (p){ log(p/(1-p))}
expit <- function(theta){ 1/(1+exp(-theta))}

set.seed(234324)
N= 5000
p1= .5
p2 = .7 

test <- multimark::simdataClosed(
  N = N,
  noccas = 2,
  pbeta = logit(c(p1, p2)),
  tau = 0, # behavioural effects (i.e. c)
  sigma2_zp = 0,
  delta_1 = 0.4,
  delta_2 = 0.4,
  alpha = 0,
  data.type = "always", # some animals have both sides read.
  link = "logit"
)

The first few encounter histories under the LRB protocol are

      [,1] [,2]
 [1,]    0    1
 [2,]    2    0
 [3,]    1    0
 [4,]    0    1
 [5,]    4    1
 [6,]    2    2
 [7,]    4    1
 [8,]    0    4
 [9,]    4    1
[10,]    2    4

We see that encounter matrix has capture histories with 1 representing a L capture and a 2 representing a R capture. A capture history with a 4 represents an animal that has both sides read.

The reduced set of capture histories is:

test.ch <- as.data.frame(test$Enc.Mat)

test.ch.red <- plyr::ddply(test.ch,c("V1","V2"), plyr::summarize, 
                           cap_hist=paste0(V1[1],V2[1]),
                           freq=length(V1))
test.ch.red

   V1 V2 cap_hist freq
1   0  1       01  993
2   0  2       02  972
3   0  4       04  351
4   1  0       10  582
5   1  1       11  275
6   1  4       14  130
7   2  0       20  596
8   2  2       22  251
9   2  4       24  131
10  4  0       40  154
11  4  1       41  150
12  4  2       42  163
13  4  4       44   76

Notice that \(\delta_B = 1-\delta_1-\delta_2 = 0.2\) so for 20% of animals encountered at a sampling event, both the left and right are photographed and the two sides can be linked. Some animals with both sides linked are also seen only on the left or right side.

Unlike before, this implies that the total number of encounters is roughly equal to that expected from the values of \(p_1\) and \(p_2\) by themselves because any animal that has both sides captures has them linked.

Number of pictures of animals at t=1  2508  vs. expected number of  2500 

Number of pictures of animals at t=1  3492  vs. expected number of  3500 

Now any animal that has both sides photographed has them linked.

We can get estimates of abundance based on the left or right photos individually in the same way as shown in this document. We note that type 4 encounters are used in both the left and right sides estimates.

For example, based on the left side photographs, the capture history matrix and frequency counts are:

   V1 V2 cap_hist freq
1   0  1       01  993
3   0  4       01  351
4   1  0       10  582
5   1  1       11  275
6   1  4       11  130
9   2  4       01  131
10  4  0       10  154
11  4  1       11  150
12  4  2       10  163
13  4  4       11   76

    n1   n2  m2   p.recap        mf
1 1530 2106 631 0.4124183 0.2996201

This gives the estimate of abundance that is consistent, but estimates of catchability that are confounded with the \(\delta\) values.

left.fit <- Petersen::LP_fit(temp, p_model=~..time)
left.est <- Petersen::LP_est(left.fit)
left.est$summary[, c("N_hat","N_hat_SE","N_hat_conf_level","N_hat_LCL","N_hat_UCL")]

     N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1 5106.467 130.4088             0.95  4857.162  5368.568

left.est$detail$data.expand[1:2,c("..time","p")]

    ..time         p
1        1 0.2996200
1.1      2 0.4124183

Similarly, the capture histories and estimated abundance based on the right side photographs is found as:

temp <- test.ch.red
temp$cap_hist <- gsub('4','2',temp$cap_hist)
temp$cap_hist <- gsub('1','0',temp$cap_hist)
temp$cap_hist <- gsub('2','1',temp$cap_hist)
temp <- temp[grepl('1', temp$cap_hist),]
temp

   V1 V2 cap_hist freq
2   0  2       01  972
3   0  4       01  351
6   1  4       01  130
7   2  0       10  596
8   2  2       11  251
9   2  4       11  131
10  4  0       10  154
11  4  1       10  150
12  4  2       11  163
13  4  4       11   76

Petersen::LP_summary_stats(temp)

    n1   n2  m2  p.recap        mf
1 1521 2074 621 0.408284 0.2994214

right.fit <- Petersen::LP_fit(temp, p_model=~..time)
right.est <- Petersen::LP_est(right.fit)
right.est$summary[, c("N_hat","N_hat_SE","N_hat_conf_level","N_hat_LCL","N_hat_UCL")]

     N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1 5079.794 131.2457             0.95  4828.962  5343.656

right.est$detail$data.expand[1:2,c("..time","p")]

    ..time         p
2        1 0.2994217
2.1      2 0.4082842

A naive estimate that combines the two estimates is a simple average with a SE assuming (incorrectly) that these estimates are independent (which is incorrect because some animals are involved in both estimates) is:

# naive simple average
naive.N.avg <- (left.est$summary$N_hat + right.est$summary$N_hat)/2
cat("Naive estimate of abundance based on simple average ", round(naive.N.avg), "\n")

Naive estimate of abundance based on simple average  5093 

naive.N.avg.se <- sqrt((left.est$summary$N_hat_SE^2 + right.est$summary$N_hat_SE^2)/4)
cat("Naive SE of average is ", round(naive.N.avg.se), "\n")

Naive SE of average is  93 

We now use multimark to estimate abundance based on the left and right photographs combined. A model that allows for different capture probabilities at each sampling event is fit by setting up a design matrix. The multimark packages does a Bayesian analysis and so you must specify the number of MCMC iterations to perform etc. The fitting function fits multiple chains in parallel using multiple cores in your machine.

# use multimark to estimate abundance

# we set up my own design matrix, because mod.p=~time creates a design matrix 
# that has 3 columns rather than only 2
myCovs <- data.frame(mytime=c(0,1))
myCovs

  mytime
1      0
2      1

max.iter <- 100000
n.thin   <- 10

multimark.res <- file.path("Images","multimark-sim-LRB.Rdata")
if( file.exists(multimark.res)){
    load(multimark.res)
}
if(!file.exists(multimark.res)){
 fit <- multimark::multimarkClosed(
  test$Enc.Mat,
  data.type = "always",
  covs = myCovs,
  mms = NULL,
  mod.p = ~mytime,
  mod.delta = ~type,
  parms = c("pbeta", "delta", "N"),
  nchains = 3,
  iter = max.iter,
  adapt = 10000,
  bin = 50,
  thin = n.thin,
  burnin = 2000)
 save(fit, file=multimark.res)
}

A summary of the MCMC output is:

summary(fit$mcmc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                         Mean        SD  Naive SE Time-series SE
pbeta[(Intercept)]   -0.01547  0.043726 2.550e-04      7.549e-04
pbeta[mytime]         0.81855  0.046360 2.704e-04      4.898e-04
N                  5057.90820 85.525972 4.988e-01      1.798e+00
delta_1               0.40083  0.006339 3.697e-05      3.697e-05
delta_2               0.39397  0.006316 3.683e-05      3.665e-05

2. Quantiles for each variable:

                        2.5%        25%        50%       75%     97.5%
pbeta[(Intercept)]   -0.1012   -0.04517   -0.01525 1.393e-02 7.082e-02
pbeta[mytime]         0.7287    0.78716    0.81828 8.493e-01 9.105e-01
N                  4895.0000 4999.00000 5057.00000 5.114e+03 5.228e+03
delta_1               0.3884    0.39657    0.40083 4.051e-01 4.133e-01
delta_2               0.3817    0.38971    0.39392 3.982e-01 4.063e-01

# get the estimates of p and c
pc <- multimark::getprobsClosed(fit, link = "logit")
summary(pc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

       Mean      SD  Naive SE Time-series SE
p[1] 0.4961 0.01093 6.372e-05      0.0001886
p[2] 0.6905 0.01335 7.787e-05      0.0002506
c[2] 0.6905 0.01335 7.787e-05      0.0002506

2. Quantiles for each variable:

       2.5%    25%    50%    75%  97.5%
p[1] 0.4747 0.4887 0.4962 0.5035 0.5177
p[2] 0.6645 0.6815 0.6904 0.6995 0.7164
c[2] 0.6645 0.6815 0.6904 0.6995 0.7164

We see that the estimate of abundance appears to be unbiased and has comparable uncertainty to our naive average of estimates of abundance based on the left or right photo individually.

But now with the LRB study design, estimates of catchability and the delta’s are not unbiased.

The MCMC diagnostic plots look fine (but are not shown here)

YMMV.

11.4 Bobcat example

The multimark package also ships with a sample dataset on a study of bobcats. Images of the left and right side were captured using camera traps in 8 sampling locations. It is not possible to match left and right photographs of the bobcats.

The first few records, with one line per photograph are:

# Get the bobcat data
data("bobcat")
head(bobcat)

    occ1 occ2 occ3 occ4 occ5 occ6 occ7 occ8
ID2    0    0    0    0    0    1    1    0
ID3    0    0    1    0    1    0    0    0
ID4    0    0    0    0    1    0    0    0
ID6    1    0    0    0    0    0    0    0
ID7    0    0    1    0    0    0    0    1
ID8    0    1    0    0    0    0    0    0

The data are very sparse.

We collapse the first 4 sampling occasions into the first “event”, and the last for sampling occasions into the second “event”. In most cases, the photo was only seen once in each of the 4 sets of occasions.

The reduced capture events are shown at the right of each line below:

# classify into 2 events; first 4 sampling events; 
# last 4 sampling events; if seen in any of the multiple occasions in an event

bobcat <- cbind(bobcat, V1=apply(bobcat[,c("occ1","occ2","occ3","occ4")],1,max))
bobcat <- cbind(bobcat, V2=apply(bobcat[,c("occ5","occ6","occ7","occ8")],1,max))
head(bobcat)

    occ1 occ2 occ3 occ4 occ5 occ6 occ7 occ8 V1 V2
ID2    0    0    0    0    0    1    1    0  0  1
ID3    0    0    1    0    1    0    0    0  1  1
ID4    0    0    0    0    1    0    0    0  0  1
ID6    1    0    0    0    0    0    0    0  1  0
ID7    0    0    1    0    0    0    0    1  1  1
ID8    0    1    0    0    0    0    0    0  1  0

This gives us a set of capture histories:

  V1 V2 cap_hist freq
1  0  1       01   11
2  0  2       02   13
3  1  0       10    6
4  1  1       11    6
5  2  0       20    6
6  2  2       22    4

The estimated abundances from the left and right sided photographs are:

Left  sided abundance estimates

     N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1 33.99992 7.895085             0.95  21.56856  53.59626

Right sided abundance estimates

  N_hat N_hat_SE N_hat_conf_level N_hat_LCL N_hat_UCL
1  42.5 14.39401             0.95  21.88275  82.54221

Naive estimate of abundance based on simple average  38.2 

Naive SE of average is  8.2 

We now use multimark on the reduced capture histories:

# we set up my own design matrix, because mod.p=~time creates a 
# design matrix that has 3 columns rather than only 2
myCovs <- data.frame(mytime=c(0,1))
#myCovs

max.iter <- 100000
n.thin   <- 10

multimark.bobcat.res <- file.path("Images","multimark-bobcat.Rdata")
if( file.exists(multimark.bobcat.res)){
    load(multimark.bobcat.res)
}
if(!file.exists(multimark.bobcat.res)){
 bobcat.fit <- multimark::multimarkClosed(
  as.matrix(bobcat[,c("V1","V2")]),
  data.type = "never",
  covs = myCovs,
  mms = NULL,
  mod.p = ~mytime,
  mod.delta = ~type,
  parms = c("pbeta", "delta", "N"),
  nchains = 3,
  iter = max.iter,
  adapt = 10000,
  bin = 50,
  thin = n.thin,
  burnin = 2000)
 save(bobcat.fit, file=multimark.bobcat.res)
}

This gives the following results

summary(bobcat.fit$mcmc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                      Mean     SD  Naive SE Time-series SE
pbeta[(Intercept)] -0.2815 0.5679 0.0033118       0.019788
pbeta[mytime]       0.8428 0.6247 0.0036431       0.016251
N                  37.5916 8.8429 0.0515727       0.092755
delta_1             0.2654 0.1382 0.0008059       0.008289
delta_2             0.2151 0.1428 0.0008328       0.008920

2. Quantiles for each variable:

                       2.5%     25%     50%      75%   97.5%
pbeta[(Intercept)] -1.34656 -0.6682 -0.2990  0.08557  0.8857
pbeta[mytime]      -0.21701  0.4203  0.7883  1.19137  2.2800
N                  26.00000 32.0000 36.0000 41.00000 60.0000
delta_1             0.05044  0.1512  0.2506  0.37023  0.5403
delta_2             0.01011  0.0916  0.1958  0.32458  0.5009

# get the estimates of p and c
pc <- multimark::getprobsClosed(bobcat.fit, link = "logit")
summary(pc)

Iterations = 2010:1e+05
Thinning interval = 10 
Number of chains = 3 
Sample size per chain = 9800 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

       Mean     SD  Naive SE Time-series SE
p[1] 0.4344 0.1305 0.0007611       0.004585
p[2] 0.6137 0.1690 0.0009855       0.007349
c[2] 0.6137 0.1690 0.0009855       0.007349

2. Quantiles for each variable:

       2.5%    25%    50%    75%  97.5%
p[1] 0.2064 0.3389 0.4258 0.5214 0.7080
p[2] 0.2975 0.4894 0.6100 0.7358 0.9357
c[2] 0.2975 0.4894 0.6100 0.7358 0.9357

The estimated abundance is comparable to our naive estimate. It is unclear how the estimates of catchability and for \(\delta\) perform (see previous section).

The diagnostic plots from the MCMC iterations are:

12 Multiple Petersen estimates

12.1 Introduction

In some cases, multiple Petersen experiments occur on study population over a short time period (when the population is assumed closed). For example, 200 fish could be initially marked as they return to their spawning grounds. At a weir, a sample of 1000 fish are taken and the number of tagged fish noted. No additional tags are applied; no tags are removed; and fish are returned to the stream. Then on the spawning ground an additional 1200 fish are sampled of which some are marked.

On the surface, this appears that a closed population model with three sampling occasions could be fit. However, this is not possible because unmarked fish captured at the weir are not tagged and returned to the population. Consequently, no capture histories can be constructed for fish first captured at the weir and released. If individually identifiable tags are used, then a capture-history could be constructed for tagged fish; but if batch marks are used, then this is also not possible because it is impossible to determine if an untagged fish is captured at the weir and then again on the spawning grounds. Finally, it sometimes occurs that additional tags are applied to some (but not all untagged fish) captured at the weir and most closed population models cannot deal with the addition of more tags partway through the study.

These types of studies are known as mark-resight models (McClintock et al., 2009; McClintock and White, 2012). There are many previous estimators for this situation - refer to McClintock and White (2012) for details - but modern approaches use the likelihood based estimators where the probability of recapture varies among the sampling occasions, the population is closed, and the number of marks available is known for each sampling occasion. McClintock et al (2012) indicated that the logit-normal estimator should only be used with sampling is without replacement so that an animal can only be recaptured a maximum of one time, and the Poisson-log-normal estimator should be used if sampling occurs with replacement, i.e., captured-fish are returned to the study population. However, the latter model will require individually-identifiable tags so that the number of times a tagged-fish is recaptured is known. If batch marks are used and the sampling fractions are small, the logit-normal estimator will provide a close approximation.

McClintock and White (2012) provide a summary of the conditions under which these two (and a third) estimators can be used in Table 33.

Table 33: Comparison of assumptions for the mark-resight estimators in Program MARK

Estimator	Geographic closure	Sampling with replacement	Known number of marks	Identifiable marks
LNE	Required	Not allowed	Required	Not required
PNE	Required	Allowed	Not required	Required
IELNE	Not required	No allowed	Required	Not required

The IELNE allows for non-closure of the population (e.g., new animals entering the population), but is not discussed here.

Note that close approximation to the LNE estimator can be obtained by pooling all of the subsequent samples (assuming that the number of marks has not changed), and treating the weir and spawning group samples like a single “larger” recapture event. Duplicate recaptures of marked or unmarked fish are treated as two separate recaptures.

There easiest way to use these models is via RMark/MARK as shown below for the LNE model. The PNE model is specified similarly, but requires individually-marked fish.

12.2 Batch marks (logit-normal model)

This model is suitable for batch marks where a capture-history cannot be determined for individual animals. It assumes that sampling is without replacement, but as long as the number of animals sampled is small relative to the population size, the estimator should perform well.

We consider a study where salmon enter a stream and are marked. Two sample are taken upstream, at a weir and on the spawning grounds as summarized in Table 34.

Table 34: Summary of data from a study with two sampling occasions and batch marks

Event	Marks available	Recapture size	Marks seen	Petersen Est
	\(n_1\)	\(n_2\)	\(m_2\)
1	200	1027	18	10,874
2	200	1305	26	9,721

We first create the capture-history file with pseudo-histories for the tagged fish ignoring the first capture event where 200 marks are applied.

data_salmon <- data.frame(
    ch=  c("10", "01",   "00"),
    freq=c( 18,   26,  200-18-26)
)
data_salmon

  ch freq
1 10   18
2 01   26
3 00  156

Notice the first two rows report the number of marks seen at each sampling event. The last history (“00”) is technically unobservable and is a “dummy” history to represent the number of marks originally applied and never seen. Again, we are assuming that the sample sizes are small relative to the population sizes that sampling without replacement is a suitable approximation. The rule needed to create the histories is that the sum of the 1’s in a column x the count must equal the number of marks seen. There are several ways in which the histories can be constructed. The above histories are more easily interpreted if a “1” is temporarily prepended to each history for the marking event.

A . (period) can be used if the number of marks varies among the sampling events (e.g. marks added between event or marks lost between events). The following capture_history file has the same number of recaptures, but 10 marked fish were lost after sampling at the weir due to the proximity of a BBQ. Consequently, the number of marks available is 200 and 190 for the two future sampling events.

data_salmon2 <- data.frame(
    ch=  c("10", "01",  "0.",  "00"),
    freq=c( 18,   26,   10,  200-18-26-10)
)
data_salmon2

  ch freq
1 10   18
2 01   26
3 0.   10
4 00  146

Now there are \(200-18-26-10=146\) marked fish not seen at both sampling occasions, with an additional 10 fish not seen at the first subsequent sampling occasion, but not available for the second subsequent sampling event. Again, prepend a “1” in front of each history to interpret the histories more readily. Histories can be constructed for adding new marks as well.

We launch RMark and see which parameters must be specified for this model:

library(RMark)

setup.parameters("LogitNormalMR", check=TRUE) 

[1] "p"     "sigma" "N"    

The parameter

• \(p\) represents the mean (logit) capture probability;
• \(sigma\) represent variation in the (logit) capture probability over individuals. Because we don’t have individually marked fish, we must set this parameter to 0 for this example.
• \(N\) represents the population abundance.

We need to use the process.data() function to specify the umber of unmarked fish seen in the experiment:

salmon.proc <- process.data(data_salmon,model="LogitNormalMR",
                    counts=list("Unmarked seen"=c(2288)),
                    time.intervals=c(0))
salmon.ddl   <- make.design.data(salmon.proc)

The time intervals of 0 indicate a single session (i.e. year) of data where the population is assume closed with two secondary sessions in this single primary session. The make.design.data() function creates the analysis structure.

Specify a model in the usual way using RMark We will start with a p(t), sigma=0 model.

Due to a ‘feature’ in MARK when called by RMark, poor initial values are chosen and the model converges to nonsense. We need to create sensible initial values on the beta scale. For the capture probabilities (the logit scale), we can use the value of 0 (once for each occasion), for the sigma parameter, we are initializing and fixing at 0 and for the abundance (N), we use log(approximate abundance). Here the approximate abundance is about 10,000 so we use log(10,000).

salmon.mt <- mark( salmon.proc,                 
         model="LogitNormalMR",
         model.name="(p(t) sigma=0)",               
         model.parameters=
               list(p     =list(formula=~time),
                     sigma=list(formula=~1, fixed=0),
                     N    =list(formula=~1)),
               initial=c(0,0,0,log(10000))       
         )

Output summary for LogitNormalMR model
Name : (p(t) sigma=0) 

Npar :  3
-2lnL:  285.0223
AICc :  291.0828

Beta
                estimate        se        lcl       ucl
p:(Intercept) -2.3127836 0.2472701 -2.7974329 -1.828134
p:time2        0.4137078 0.3248364 -0.2229716  1.050387
N:(Intercept)  9.2480824 0.1432740  8.9672654  9.528899


Real Parameter p
 Session:1 
         1         2
 0.0900698 0.1302131


Real Parameter sigma
 Session:1 
  0
 NA


Real Parameter N
 Session:1 
         
 10584.63

salmon.mt$results$real  

                      estimate           se          lcl          ucl fixed
p g1 s1 t1        9.006980e-02    0.0202656    0.0574631 1.384607e-01      
p g1 s1 t2        1.302131e-01    0.0238270    0.0901817 1.844126e-01      
N g1 s1           1.058463e+04 1487.8476000 8053.3129000 1.393186e+04      
sigma g1 a0 s1 t0 0.000000e+00    0.0000000    0.0000000 0.000000e+00 Fixed
                  note
p g1 s1 t1            
p g1 s1 t2            
N g1 s1               
sigma g1 a0 s1 t0     

The estimated abundance is

get.real(salmon.mt, "N", se=TRUE)

        all.diff.index par.index estimate       se      lcl      ucl fixed note
N g1 s1              4         3 10584.63 1487.848 8053.313 13931.86           
        group session
N g1 s1     1       1

We can fit a model where the capture probabilities are equal across both subsequent sampling events:

salmon.m0 <- mark( salmon.proc,                
        model="LogitNormalMR",
        model.name="(p(.) sigma=0)",               
        model.parameters=
             list(p    =list(formula=~1),
                  sigma=list(formula=~1, fixed=0),
                  N    =list(formula=~1)),
         initial=c(0,0,log(10000))   )

Output summary for LogitNormalMR model
Name : (p(.) sigma=0) 

Npar :  2
-2lnL:  286.6689
AICc :  290.699

Beta
               estimate        se       lcl       ucl
p:(Intercept) -2.089332 0.1598152 -2.402570 -1.776095
N:(Intercept)  9.248113 0.1435710  8.966714  9.529512


Real Parameter p
 Session:1 
        1        2
 0.110138 0.110138


Real Parameter sigma
 Session:1 
  0
 NA


Real Parameter N
 Session:1 
         
 10584.95

salmon.m0$results$real

                      estimate           se          lcl          ucl fixed
p g1 s1 t1            0.110138    0.0156631    0.0829769     0.144786      
N g1 s1           10584.951000 1490.9778000 8049.0515000 13940.158000      
sigma g1 a0 s1 t0     0.000000    0.0000000    0.0000000     0.000000 Fixed
                  note
p g1 s1 t1            
N g1 s1               
sigma g1 a0 s1 t0     

The model comparison table is constructed in the usual way.

salmon.modset <- collect.models( type="LogitNormalMR")
salmon.modset

           model npar     AICc DeltaAICc    weight Deviance
1 (p(.) sigma=0)    2 290.6990 0.0000000 0.5478212 286.6689
2 (p(t) sigma=0)    3 291.0828 0.3837426 0.4521788 285.0223

You CANNOT use the standard model.average() function due to the way MARK and RMark interact (groan)! Notice that the model averaged estimate is \(N\) is WRONG.

salmonavg.N.wrong <- model.average(salmon.modset, parameter="N", vcv=TRUE) # WRONG ANSWER 
salmonavg.N.wrong

$estimates
        par.index estimate       se      lcl      ucl fixed note group session
N g1 s1         4 10384.81 1489.563 7839.795 13755.99                1       1

$vcv.real
        4
4 2218798

The model averaged value of \(N\) is lower than each of the estimates which is impossible because it actually extracts the estimated UNMARKED population abundance from each subsequent capture event (groan).

You need to create a list structure to use model.average.list() function – see the help file on this function.

est.list <- function(modset){
    # Extract the estimate, AICc, and SE from the model sets
    # we need list with three vectors
    estimate <- plyr::laply(modset[ names(modset) != "model.table"], 
                            function(x){get.real(x, "N", se=TRUE)$estimate})
    weight   <- modset$model.table$weight
    se       <- plyr::laply(modset[ names(modset) != "model.table"], 
                            function(x){get.real(x, "N", se=TRUE)$se})
    list(estimate=estimate, weight=weight, se=se)
}
est.to.average <- est.list(salmon.modset)
est.to.average

$estimate
[1] 10584.95 10584.63

$weight
[1] 0.5478212 0.4521788

$se
[1] 1490.978 1487.848

This creates a list with 3 vectors corresponding to the estimates to be averaged, the weights for the estimates, and the SE of the individual model estimates.

Finally, we can model average using this data structure (groan):

salmonavg.N <- model.average(est.to.average, mata=TRUE)  # mata argument request ci's
salmonavg.N

$estimate
[1] 10584.81

$se
[1] 1489.563

$lcl
[1] 7665.316

$ucl
[1] 13504.3

[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

This procedure is not for the faint of heart!

• Capture histories are “imaginary” and are devices to getting the information into MARK. In particular the use of the 00 and 0. histories to allow for marks to be added and removed is not intuitive.
• The total count of unmarked animals is specified in the process.data() function.
• Need to specify good starting values on the beta (logit or log scale depending on the parameter) scale (groan)!
• Cannot use the standard model.average() function and need to use the model.average.list() function (groan).

As a point of interest, the pairs of Petersen estimates formed at the second and third sampling event and by pooling the subsequent capture events are shown in Table 35.

Table 35: Mean of the Petersen estimator and Petersen estimator form by pooling all samples

Events	Estimate	SE
1 & 2	10,874	2292
1 & 3	9,721	1692
Average	10,297	????
Pooled	10,420	1340

12.3 Example of PNE method - TBA

12.4 Summary

13 Forward and reverse capture-recapture studies

13.1 Introduction

Consider a capture-recapture study of fish returning to spawn that consists of many distinct stocks with different spawning grounds.

In a forward capture-recapture recapture study, fish are captured at the entrance to the river, tagged and released. Then on one particular spawning ground, fish are examined for tags. But in many cases, biological samples are taken are also taken when fish are tagged, and genetic methods used to estimate the stock proportions.

This can be used in “reverse” capture-recapture study. If fish are fully enumerated on the spawning ground (e.g., at a fence), this is considered the “tagging” event. Then working backward, the biological samples are the “recapture” event.

This methodology is explained in more detail in Hamazaki and DeCovich (2014). The key assumptions for the genetic reverse-capture-recapture method (in addition to the usual for a Petersen estimator) include:

1. Complete enumeration of stock on the spawning ground or at a weir. A complete enumeration of the stock is needed to ensure that the “tagging” event is complete. This assumption could be violated by the stock spawning elsewhere (e.g., main stem), or not all spawning areas surveyed. Or multiple stocks with different genetic baselines could spawn in the same spawning grounds.
2. The genetic baseline is complete and accurate. The baseline needs to be complete in-order to confidently assign individuals to the “marked” (genetically distinct) population. If incomplete, individuals from genetically similar populations may be assigned to the “marked” population (see below) because it is the “closest” populations for assignment. This assumption can be relaxed if “not in baseline” is an acceptable assignment for stocks not in the baseline. The genetic baseline must also be accurate, e.g., fish the form the baseline really belong to the stock of interest and are not a mixture of multiple stocks.
3. Marked population is genetically distinct. The identified “marked” population needs to be genetically distinct from all other populations in the system so that individuals from genetically similar populations are not assigned to the “marked” population introducing bias in the estimated proportion of the “marked” population during the “second” sampling occasion at the entrance to the system.
4. Marked population size. While the size of the “marked population” theoretically has no bearing on the estimator, a small stock may have too few “recaptures” in the biological samples and so the estimator may have very poor precision.
5. Equal en route mortality among different components of the run. En route morality in the reverse capture-recapture method acts like “immigration” in the forward capture-recapture methods and so leads to valid estimates at the entrance to the system. Homogeneous en route mortality may be violated when there are differential stock-specific harvest rates among stocks. This assumption can be assessed in the harvest through rigorous tissue collection of harvested individuals. The assumption may also be violated if stock have different distances between the entrance to the system and their spawning sites and so the en route mortality may also differ. One way to test this assumption is via the goodness of fit testing of equal recovery probabilities discussed earlier.
6. Random sampling at the entry to the system. Random sampling at the entry to the system could occur if, for example, fishwheels are sampling approximately proportional to the run and a constant fraction of the tagged fish have biological samples taken, e.g., every nth tagged fish is sampled.

13.2 Example - Run size of Yukon River Chinook

This is the 2011 data from Hamazaki and DeCovich (2014) taken from Table 2 of their paper.

Briefly, Chinook Salmon returning to Yukon River are counted in the lower river (Pilot Station) using hydroacoustic and sonar methods. At the same time, biological samples are taken throughout the run and the stock of origin was determined using GSI methods. In particular, in 2011, the proportion of stocks of Canadian origin was 0.35 (SE .030) based on a sample size of 251 fish of which 87 were fish of Canadian origin.

Fish continue their upward migration and the number of fish that enter Canadian waters was counted using sonar at the Eagle Border Station (51,271 (SE 135)). To this was added the estimated harvest of Canadian fish between Pilot Station and Eagle Border Station from harvest monitoring given a total of 66,225 (SE 1514) Canadian fish. These are considered to be the “tagged” fish.

Now consider running the experiment in reverse. We have 66,225 are marked at Eagle Border and swim “backwards” to Pilot Station. Here a biological sample of 251 fish were extracted, of which 87 were fish from Canada. This is the “recapture” sample. So, in reverse, we have \(n_1=66,225\); \(n_2=251\), \(m_2=87\) giving a reverse Petersen estimate of \[\widehat{N}_{reverse}=\frac{n_1 \times n_2}{m_2}=\frac{66,225 \times 251}{87}=191,062\] as reported in their paper. [Their estimate is slightly different because of rounding.]

The data are available in the Petersen package:

data(data_yukon_reverse)
data_yukon_reverse

  cap_hist  freq         se
1       10 66138 1574.00000
2       11    87    7.53865
3       01   164    0.00000
                                                     comment
1 Escapement + harvest - genetic samples recaptured cdn fish
2                           Genetic sample that are cdn fish
3                       Genetic sample that are non-cdn fish

The estimates are obtained in the usual way:

yukon.fit <- Petersen::LP_fit(data_yukon_reverse,
                                p_model=~..time)
yukon.est <- Petersen::LP_est(yukon.fit, N_hat=~1)
yukon.est$summary

  N_hat_f    N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      ~1 (Intercept) 191062.9 16546.88             0.95              logN
  N_hat_LCL N_hat_UCL p_model name_model   cond.ll n.parms  nobs  method
1  161234.7  226409.2 ~..time p: ~..time -1812.538       2 66389 CondLik

The reported SE is too small because it does not account for the uncertainty in the \(n_1\) (uncertainty in harvest and from sonar) and \(m_2\) (because genetic assignment has some error). The LP_est_adjust() function can again be used. First, we get the adjustment factors for \(n_1\) and \(m_2\) based on the reported uncertainties:

n1.adjust.est <- 1
n1.adjust.se  <- data_yukon_reverse$se[ data_yukon_reverse$cap_hist=="10"] / data_yukon_reverse$freq[ data_yukon_reverse$cap_hist=="10"]
cat("Estimated adjustment factors for n1 ", n1.adjust.est, n1.adjust.se, "\n")

Estimated adjustment factors for n1  1 0.02379872 

m2.adjust.est <- 1
m2.adjust.se  <- data_yukon_reverse$se[ data_yukon_reverse$cap_hist=="11"] / sum(data_yukon_reverse$freq[ data_yukon_reverse$cap_hist!="10"])
cat("Estimated adjustment factors for m2 ", m2.adjust.est, m2.adjust.se, "\n")

Estimated adjustment factors for m2  1 0.03003446 

And, now the adjustment factors are applied

set.seed(34543534)
yukon.adjust <- LP_est_adjust(
              yukon.est$summary$N_hat, yukon.est$summary$N_hat_SE,
              n1.adjust.est=n1.adjust.est, n1.adjust.se=n1.adjust.se,
              m2.adjust.est=m2.adjust.est, m2.adjust.se=m2.adjust.se
              )
yukon.adjust$summary

  N_hat_un N_hat_un_SE N_hat_adj N_hat_adj_SE N_hat_adj_LCL N_hat_adj_UCL
1 191062.9    16546.88  191996.4     18417.44      158396.5        230678

The final answer matches those reported in the paper (\(\widehat{N}_{paper}=191,155\) (SE \(18,372\))) where the authors used a bootstrapping procedure (similar to that used in the LP_est_adjust() function).

13.3 Example: Lower Fraser Coho - SPAS in reverse

This example was provided by Kaitlyn Dionne from the Department of Fisheries and Oceans, Canada and is discussed in Arbeider et al (2020) in more detail.

Arbeider et al (2020) proposed to estimate the run size of Lower Fraser River Coho (LFC) using a geographically stratified reverse-capture method. Briefly, as LFC coho swim upstream in the Fraser River, they are sampled near New Westminister, BC, which is downstream from several major rivers with large spawning populations. These sampled fish are assigned to the spawning population using genetic stock identification and other methods. These spawning populations are identified as the Chilliwack Hatchery (denoted C), the Lilloet River natural spawning population (denoted L), the Nicomen Slough population (denote N) and all other populations (denoted as 0). Notice that the sampled fish at New Westminister are NOT physically tagged, and population assignment is through genetic stock identification and other measures.

The upstream migration extends over two months (September to October) and is divided into 3 temporal strata corresponding to Early (denoted 1E), Peak (denoted 2P) and Late (denoted 3L). The digits 1, 2, 3 in front of the codes ensures that the temporal strata are sorted temporally, but this is merely a convenience and does not affect the results.

The spawning populations at C, L, and N are estimated by a variety of methods (see Arbeider, et al. 2020). Each of the population estimates also has an estimated SE (which will be ignored for now).

The data are available in the Petersen package:

data(data_lfc_reverse)
data_lfc_reverse

   cap_hist  freq    SE
1     C..1E    23    NA
2     C..2P    44    NA
3     C..3L     6    NA
4      C..0 66127 13039
5     L..1E    12    NA
6     L..2P     8    NA
7     L..3L     2    NA
8      L..0 14300   403
9     N..1E     1    NA
10    N..2P    13    NA
11    N..3L     3    NA
12     N..0 15274   461
13    0..1E    37    NA
14    0..2P   146    NA
15    0..3L    46    NA

This can be displayed in a matrix form as:

   s2
s1      0    1E    2P    3L
  0     0    37   146    46
  C 66127    23    44     6
  L 14300    12     8     2
  N 15274     1    13     3

The first row (corresponding to s1=0) are the number of fish in the New Westminister sample assigned to stocks other than C, L, or N. The first column (corresponding to s2=0) are the estimated total escapement in each of C, L, N (less the fish removed for genetic identification which are trivially small). The matrix in the lower bottom of the above array shows the number of fish assigned from the New Westminister sample to the four (C, L, N, and 0) geographic strata in each of the 3 time periods (1E, 2P, or 3L).

Notice that this data set up is going in REVERSE from total spawning escapement backwards to the genetic sample in New Westminister.

13.3.1 Fitting the SPAS model

The SPAS model is fit to the above data using the wrapper supplied with this package. We start with no pooling of rows or columns.

mod..1.fit <- Petersen::LP_SPAS_fit(data_lfc_reverse,
                       model.id="LFC - entire matrix",
                       row.pool.in=1:3, col.pool.in=1:3,quietly=TRUE)

The summary of the fit is:

mod..1.fit$summary

                     p_model          name_model  cond.ll n.parms  nobs method
1 Refer to row.pool/col.pool LFC - entire matrix 923265.5      15 96042   SPAS
  cond.factor
1    2095.344

The usual conditional likelihood is presented etc. Of more importance is the condition factor. This indicates how close to singularity the recovery matrix (in this case, the stock assignment of New Westminister fish to the geographic and temporal strata) is with larger values indicating that the matrix of “recoveries” is closer to singularity. Usually, this value should be 1000 or less to avoid numerical issues in the fit. Here the conditional factor is around 2000 which is a bit concerning but still likely to be acceptable.

We also fit a model (not shown) where the first two geographic strata (C and L) were pooled and give essentially the same abundance estimate.

We estimate the total LFC population:

mod..1.est <- Petersen::LP_SPAS_est(mod..1.fit)
mod..1.est$summary

  N_hat_f N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 289861.2 42516.79             0.95      Large sample
  N_hat_LCL N_hat_UCL                    p_model          name_model  cond.ll
1  206529.8  373192.6 Refer to row.pool/col.pool LFC - entire matrix 923265.5
  n.parms  nobs method cond.factor
1      15 96042   SPAS    2095.344

The reverse capture-recapture stratified-Petersen estimate of the total number of fish passing the New Westminister sampling station is 289,861 with a standard error of 42,517.

The pooled-Petersen estimator can be obtained by pooling all rows to a single row:

mod..PP.fit <- Petersen::LP_SPAS_fit(data_lfc_reverse, model.id="Pooled Petersen",
                       row.pool.in=rep(1,3),
                       col.pool.in=c(1,2,3),quietly=TRUE)
mod..PP.fit$summary

                     p_model      name_model  cond.ll n.parms  nobs method
1 Refer to row.pool/col.pool Pooled Petersen 923258.6      13 96042   SPAS
  cond.factor
1           1

mod..PP.est <- Petersen::LP_SPAS_est(mod..PP.fit)
mod..PP.est$summary

  N_hat_f N_hat_rn    N_hat N_hat_SE N_hat_conf_level N_hat_conf_method
1      NA       NA 291716.4 22575.55             0.95      Large sample
  N_hat_LCL N_hat_UCL                    p_model      name_model  cond.ll
1  247469.1  335963.6 Refer to row.pool/col.pool Pooled Petersen 923258.6
  n.parms  nobs method cond.factor
1      13 96042   SPAS           1

The capture-recapture reverse pooled-Petersen estimate of the total number of fish passing the New Westminister sampling station is 291,716 with a standard error of 22,576, which is virtually identical to the reversed stratified estimate but with a considerably reduced SE.

The above analysis ignored the uncertainty in the estimated escapement and so the reported standard errors for abundance are under reported. An approximate measure of this additional uncertainty for the pooled-Petersen reverse capture-recapture estimate could be obtained by bootstrapping, or using the LP_est_adjust() function.

# get the total of "n1" (reverse) and its approximate SE
temp <- data_lfc_reverse[ grepl("0$", data_lfc_reverse$cap_hist),]
n1 <- sum(temp$freq)
n1.rse <- sqrt(sum(temp$SE^2))/n1

est.adj <- LP_est_adjust(mod..PP.est$summary$N_hat, mod..PP.est$summary$N_hat_SE, n1.adjust.est=1, n1.adjust.se=n1.rse)
est.adj$summary

  N_hat_un N_hat_un_SE N_hat_adj N_hat_adj_SE N_hat_adj_LCL N_hat_adj_UCL
1 291716.4    22575.55  292894.4     46348.98      205007.6      388175.5

This shows that the uncertainty in the escapement in the terminal run would increase the SE of the abundance estimates by about 50%!

13.4 Example combining forward and reverse capture-recapture studies

TBA

13.5 Summary

The reverse capture-recapture method requires no mortality (either natural or harvest) between the location of the biological sample and the final spawning grounds. Otherwise, these sources of removal must be added to the “tagged” group when applied in reverse.

This is different from the forward capture-recapture experiment where mortality between the tagging and recapture events is allowable if it applies equally to tagged and untagged fish. When running the estimator in reverse, it would appear on first glance that any mortality should have similar effects to immigration where the abundance could still be estimated at the second sampling event. However, the key difference is that immigration in the forward direction applies only to untagged fish, but in the reverse direction mortality (now appearing to be immigration) is being applied to “tagged” fish.

14 Genetic capture-recapture methods using close kin studies

Genetic markers that unique identify individual can used in place of marks or tags and pose no unique challenges. An interesting new application is the use of transgenerational marks where the set of animals sampled are different in the two events.

Single sample methods? Like the sequential capture-recapture methods taking one sample at a time? Rarefaction curves?

14.1 Transgenerational capture-recapture

14.1.1 Introduction

In these studies, The first sample is of adults is captured and a set of genetic markers (e.g., SNPs) are established for each captured animals. These adults mate and reproduce. A second sample of offspring is captured. The genetic information from these offspring is obtained. If the genetic markers are sufficiently diverse, then a parent-offspring-pair may be established in some cases, i.e., for some of the offspring, it can be established if the father and/or mother arose from the initial marked set (Figure 27).

Figure 27: Schematic of transgenerational capture-recapture. Solid outline indicates a sampled fish. For juvenile fish, both the mother (left) and father (right) fish id are determined. Shaded areas indicate fish ids from genetic markers in juveniles that match genotyped adult fish. Notice that some fish (e.g. fish 8) did not produce offspring in the juvenile population. Mothers and fathers are not monogamous.

Notice that the set of animals sampled at the two sample events is different. Each juvenile sampled has two “genetic tags” (one for the father and one for the mother). Because a parental pair can give rise to many offspring and mother/father relationships may not be monogamous, multiple juveniles can have the “genetic tag” for a particular adults fish in the original samples.

Rawding et al (2014) used this method to estimate the number of returning Chinook salmon in the Coweeman River, Washington State. The marks were the genotyped carcasses collected from the spawning area during the first sampling event. The second sampling event consisted of a collection of juveniles from a downstream migrant trap located below the spawning area. The parents that assigned to the juveniles through parentage analysis were considered the recaptures, which was a subset of the genotypes captured in the second sample.

Rosenbaum et al (2024) also used a similar method to estimate the number of returning Chinook salmon in the Chilkat River in Alaska State. Returning adult Chinook salmon were captured during freshwater migration in June and July using fishwheels and drift gillnets or in August using rod and reel, dip nets, short tangle nets, beach seines, and carcass surveys. Juvenile tissue samples were collected in the fall of 2021 using a stratified systematic sampling design, with samples collected continuously throughout September and October, across the mainstem and two of the three primary spawning tributaries using baited minnow traps.

In both examples, kinship relationships among adults and juveniles was determined using the parentage analysis program COLONY (Wang & Santure, 2009).

COLONY is a full probability pedigree reconstruction software that uses maximum likelihood to reconstruct full- and half-sibling family groups among juveniles and assigns parents to family groups. Additionally, COLONY infers genotypes of unsampled parents through information of sibling relationships among juveniles. Reconstructing unsampled parental genotypes allows for putative identification of the total number of reproductively successful adults, both sampled and unsampled.

14.1.2 Bailey Binomial model

In the first model, the genetically typed adults are treated as the marked sample, and each juvenile genetically sampled is treated as two (independent) draws from the population of adults. Because a parental pair can produce multiple offspring, the juvenile samples treated as sampling with replacement. These are the conditions suitable for a Bailey (1951, 1952) binomial estimator.

Rosenbaum et al (2024) provided the raw genetic matching data from their study. When processed, this gives rise to the summary data in Table 36:

Table 36: Summary data from transgenerational capture-recapture study on Chilkat River, Alaska.

Capture history	Total captures	Frequency
01	1	279
01	2	191
01	3	86
01	4	21
01	5	9
01	6	6
01	7	5
01	9	1
10	0	434
11	1	96
11	2	31
11	3	11
11	4	5
11	5	1
11	6	1
11	7	2

A total of \(M=\) 581 adults were captured and genotyped (histories 10 and 11 in Table 36). Most were not “recaptured” by the father/mother genotyping of the juveniles, but 96 adults were “recaptured” in the juvenile sample once, 31 adults were captured 2x; etc.

A total of 682 juveniles were genotypes and each gives rise to 2 parental genotypes, some of which are matched, amd 279 juvenile mother/father slots were not recaptures of the adults from previous fall. This gives \(C=\) 1364 (number of juvenile genotypes measured - two from each fish).

Finally, a total of \(R =\) 236 “recaptures” (including double, triple counting etc.) were found.

The (bias adjusted) Bailey Binomial estimate of adults abundance in the fall is \[\widehat{N}_{BB} = \frac{M \times (C+1)}{R+1}\] and found to be 3346 (SE 197) fish.

14.1.3 Hypergeogeometric model

The Bailey Binomial model assumes that on each draw (i.e., each juvenile) has an equal chance of selecting each adult parental genotype. However, a more fecund fish will likely give rise to more juveniles, and so the assumption of equal probability of sampling an adults fish is violated.

For this reason, the number of UNIQUE genotype in the juvenile sample can be used in the usual hypergeometric (Petersen or Chapman) estimator.

The value of \(M\) (number adults genotypes) is unchanged. The number of unique adults recaptured can be derived from Table 36 by treating multiple recaptures of the same adult genotype as single capture event. This give \(R_{unique}\) = 147.

Similarly, the number of unique adult genotypes seen for the first time is also determined from Table 36. For this study this gives \(C_{unique}\) = 745.

This gives an estimate of 2933 (SE 186) fish.

Both estimates are similar.

14.1.4 Caution on using simple sample methods.

It might be tempting to use single-sample methods based on the sample of juveniles alone such as species rarefaction curves, or methods developed for capture-recapture that allow for heterogeneity (such as Chao’s estimate for the lower bound on the number of species). However, these methods will give you an estimate for the number of adults that successfully bred and had juveniles that suvived to the time of sampling. This can be substantially less than the total number of adults that returned to spawn.

For example, Chao’s simple estimator of \[\widehat{N}_{Chao} = C_{unique} + \frac{f_1^2}{2 f_2}\] where \(f_i\) is the number of adult genetic codes captured \(i\) time in the juvenile sample, and \(C_{unique}\) is the number of unique adult genetic codes seen in the juvenile sample gives an estimate of 949 adults, which is substantially less than the other estimates, presumably because of the large number of returning adults who do not breed, nor had juveniles that survived to the time of sampling.

14.1.5 Limitations and Assumptions

The key assumptions needed to ensure that the Petersen (and related) estimators are unbiased are

1. there is no mark loss,
2. there are no marking effects,
3. the population is closed,
4. all marked and unmarked fish are correctly identified and enumerated, and
5. all fish in the population have an equal probability of capture in at least one sample event, or there is complete mixing of marked and unmarked individuals. There are additional, biologically unlikely, ways.

Genetic-based approaches meet the no tag-loss and no marking-effects assumptions, because genotypes were permanent markers and genetic marks were obtained from adults or from juveniles.

The closure assumption requires that within the study period there are no additions to the population through births or immigration and no deletions through death or emigration. In this studies, the closure assumption is likely violated by adults failing to produce offspring; however, the Petersen estimator is unbiased with respect to the population at time of tagging if mortality was random. Similarly, if the carcass collection was unbiased regarding the production of at least one offspring and the mortality (i.e., lack of spawning success) was equal for sampled and unsampled carcasses. There is likely no a priori reason to believe carcass sampling was not representative relative to the production of at least one offspring.

Regarding juveniles, careful study design is required to ensure that only juveniles from the adults population in the fall are sampled.

Good laboratory procedures and sufficient genetic markers are needed to ensure that correct identification and reporting of tagged or marked fish (assumption 4).

The last assumption (homogeneity of capture probabilities in at least one sample event) is the most difficult to ensure is being met. A representative carcass sampling design can be used to obtain a sample of parents. For tGMR, differences in individual parental reproductive success create unequal capture probabilities for parents in the second sampling event. When repeatedly sampling the same individual, the heterogeneity in capture probabilities can be partially alleviated by using only unique adults captures.

Equal mixing would imply that juveniles produced by the adults marked fish mix completely with the juveniles produced by unmarked adult fish. This could be violated by geographic separation of spawning adults, etc.

Rosenbaum et al (2024) concluded that the binomial tGMR estimate is less robust to assumption violations and more prone to biases as a result of following sampling with replacement. They caution tGMR users from relying on the binomial estimator. Instead, it is preferable to identify an informative panel of genomic markers for the target population to confidently infer unsampled parents during parentage analysis so that the hypergeometric framework may be reliably utilized.

Heterogeneity can be (partially) alleviated through the use of covariates as presented early in this manuscript. However, covariates are readily available for the “marked” sample of adults, but not for the adults never seen before as identified by the juvenile sampling. It may be possible to identify sex by a marker, but other geographic or temporal covariates cannot be assigned to every unique adult seen. This is a major limitation of the use of genetic methods in capture-recapture experiments.

Rosenbaum et al (2024) also present a simulation program to examine the impact of various assumption violations on the estimates. Consult their paper for more details.

Rawding et al (2014) also conclude

The genetic methods we describe can be used with other anadromous salmonids that immigrate to the ocean shortly after emergence, such as Chum Salmon Oncorhynchus keta and Pink Salmon O. gorbuscha. Yet, additional considerations are needed to extend this application to anadromous salmonids with yearling life histories, such as spring Chinook Salmon, Coho Salmon O. kisutch, and Atlantic Salmon Salmo salar. Further, Atlantic Salmon and steelhead O. mykiss produce unique challenges because smolts may be the offspring of anadromous fish or resident fish or both. Using our methods, the Nc and Nb for steelhead would include the combined resident and anadromous spawning population, which may not meet manager needs, as the anadromous life history form is listed for protection under the ESA, while individuals exhibiting a resident life history are not listed.

The tGMR approach could also be extended to estimate adult abundance using only returning adults. Rather than the second sampling event consisting of juvenile captures, the second event would be a capture of returning adults. By aging the scales from returning adults, the adults could be assigned to the appropriate brood year based on scale ages, and tGMR could then be used to estimate spawner abundance for relevant brood years. Another variant of the genetic mark–recapture design is to estimate smolt abundance via “back-calculation” (Volkhardt et al. 2007). In this approach the smolts are the marks and the returning adults are the recaptures and captures.

15 Grand summary

The Petersen estimator is the simplest of capture-recapture studies but should not be analyzed using simplistic methods.

• Lessons from Petersen estimator area transferable to other experiments and models
• Use conditional MLE because it can deal with covariates in a unified fashion.
• Chapman correction factor only needed for small samples, but with small numbers of marks back, estimate will have poor precision.
• Confidence intervals can be found in many ways:
  • \(\widehat{N} \pm 2 se\) in large samples
  • Transform to \(log()\) scale, find ci, back transform in smallish samples

The key assumptions are:

• Population is closed
  • If death only, then estimates \(\widehat{N}_1\)
  • If births only, then estimates \(\widehat{N}_2\)
  • If births and deaths, then estimates total animals available
• No tag loss
  • Leads to positive bias in estimated population size
  • Need to double tag some or all of animals with permanent mark or second tag of same or different type
  • Need to adjust SE if only an estimate of tag loss is available.
• All tags seen and/or all tags reported
  • Leads to positive bias in estimate population size
  • Resample to look for lost tags; offer reward tags
  • Need to adjust SE if only estimate of tag reporting is available.
• No marking effect on catchability
  • Trap happiness leads to negative bias in population estimate
  • Trap shyness leads to positive bias in population estimate
  • Impossible to design study to adjust for problem as no control group possible
  • Perhaps use different trapping methods
• No marking effect on survival (e.g. no acute tagging mortality)
  • Leads to positive bias in population estimate
  • Impossible to design study to adjust for problem as no control group possible
• Complete Mixing and/or homogeneity of catchability
  • Pure heterogeneity leads to negative bias in population estimate
  • Mixed heterogeneity can lead to positive or negative bias in population estimate
  • Good study design to ensure that all of relevant population is sampled; gear should be non-selective
  • Regular, or Geographical (SPAS) or temporal stratification (BTSPAS) may be needed.

You are likely to unsatisfied with geographical stratification because of the many parameters that must be estimated and with sparse data, the high degree of singularity in the recapture matrix.

It is essential to recapture sufficient number of marked animals.

• Rule of thumb is that \(rse \approx \frac{1}{\sqrt{marks~returned}}\)
• Precision can be improved by combining multiple Petersen estimates
• Possible to combine Petersen with other methods such as acoustic sampling; change in ratio; CPUE; etc (see me for details)

No amount of statistical wizardry can salvage a poorly designed study with insufficient marks back.

16 Miscellaneous notes

16.1 Speeding computation times

The computations in the examples in this monograph proceed quite quickly. However, there are times where the individual capture-history data structure may involve many thousands of individuals all with same capture-history.

Computation times can be considerably reduced by creating a condensed data set where the freq variable represents the number of animals with the same capture history. This speed up can be important when doing simulation studies.

data(data_NorthernPike)

Consider the NorthernPike example. This has rows for the 7805 individual fish in the capture-history array. This can be condensed to a similar format at the Rodli example:

# create condensed dataset
data_NorthernPike.condensed <- plyr::ddply(data_NorthernPike, 
                                      c("cap_hist","Sex"), 
                                      plyr::summarize,
                                      freq=length(Sex))
data_NorthernPike.condensed
##   cap_hist Sex freq
## 1       01   F  524
## 2       01   M  459
## 3       10   F 3956
## 4       10   M 2709
## 5       11   F   89
## 6       11   M   68

Both data forms give the same estimates (as expected) but the difference in computation times is:

# Fit the various models
t.condensed.start <- Sys.time()
nop.condensed     <- Petersen::LP_fit(data_NorthernPike.condensed, p_model=~-1+..time)
t.condensed.end   <- Sys.time()

t.full.start <- Sys.time()
nop.full     <- Petersen::LP_fit(data_NorthernPike,           p_model=~-1+..time)
t.full.end   <- Sys.time()

cat("The two estimates \n")
## The two estimates
nop.condensed$summary
##        p_model      name_model   cond.ll n.parms nobs  method
## 1 ~-1 + ..time p: ~-1 + ..time -3702.341       2 7805 cond ll
nop.full     $summary
##        p_model      name_model   cond.ll n.parms nobs  method
## 1 ~-1 + ..time p: ~-1 + ..time -3702.341       2 7805 cond ll

cat("Time to run the condensed data structure: ", t.condensed.end - t.condensed.start, "seconds \n")
## Time to run the condensed data structure:  0.0771389 seconds
cat("Time to run the full      data structure: ", t.full.end      - t.full.start,      "seconds \n")
## Time to run the full      data structure:  0.991904 seconds

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18 Appendix A - The multinomial models for the Petersen study

As indicated earlier, the predominant paradigm currently used in capture-recapture studies is the multinomial model based on the observed capture histories. This formulation assume that neither \(n_1\) nor \(n_2\) nor \(m_2\) are fixed quantities, but are all random variables.

There are two (asymptotically) equivalent multinomial formulations. The first is the complete multinomial where the abundance enters directly into the likelihood formulation. In this formulation an implicit assumption is that capture probabilities are known for animals never seen, i.e., are equal to the capture probabilities for animals that are seen in the study. In some cases, catchability is heterogeneous among fish, and the catchability may be related to a fixed covariates, such as fish length. Unfortunately, the fish length for fish never captured is unknown and so the capture probability for fish with history (00) cannot be modeled. Huggins (1989) showed that a conditional (upon fish been seen) multinomial model could be used in these circumstances. This chapter will develop these two formulations.

18.1 Complete multinomial model

The statistics for this formulation are the number with each capture history, \(n_{10}, n_{01}, n_{11}\). Assuming no losses on capture, the probabilities of these histories and history (00) are

\[P\left( {\left\{ {10} \right\}} \right) = p_1 \left( {1 - p_2 } \right) \] \[P\left( {\left\{ {01} \right\}} \right) = \left( {1 - p_1 } \right) p_2\] \[P\left( {\left\{ {11} \right\}} \right) = p_1 p_2\] \[P\left( {\left\{ {00} \right\}} \right) = \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)\]

The likelihood function is a multinomial distribution among these observed capture histories and probabilities: \[L= \binom{N}{n_{00},n_{01},n_{10},n_{11}} \times \left[ {p_1 \left( {1 - p_2 } \right)} \right]^{n_{10} } \left[ {\left( {1 - p_1 } \right)p_2 } \right]^{n_{01} } \left[ {p_1 p_2 } \right]^{n_{11} } \left[ {\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)} \right]^{N - n_{10} - n_{01} - n_{11} }\] The key difference between ordinary multinomial distributions and this likelihood is the presence of the unknown parameter \(N\) in the combinatorial term.

The estimators are found by maximizing the \(log(L)\) by taking first-derivatives, setting these to zero, and solving. The estimating equations for \(p_1\) and \(p_2\) are found to be:

• \(\left( {n_{11} + n_{10} } \right) = Np_1\)
• \(\left( {n_{11} + n_{01} } \right) = Np_2\)

Because the parameter \(N\) appears in factorial term, rather than taking first derivatives, the first difference is found, i.e., \(log(L(N))-log(L(N-1))=0\) which leads to the estimating equation: \[\left( {\frac{{N\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{N - n_{10} - n_{01} - n_{11} }}} \right) = 0\] Not surprisingly, this leads to the same estimators as seen earlier, namely: \[\hat N = \frac{{\left( {n_{11} + n_{10} } \right)\left( {n_{11} + n_{01} } \right)}}{{n_{11} }} = \frac{{n_1 n_2 }}{{m_2 }} \] \[\hat p_1 = \frac{{\left( {n_{11} + n_{10} } \right)}}{{\hat N}} = \frac{{n_{11} }}{{\left( {n_{11} + n_{01} } \right)}} = \frac{{m_2 }}{{n_2 }}\] \[\hat p_2 = \frac{{\left( {n_{11} + n_{01} } \right)}}{{\hat N}} = \frac{{n_{11} }}{{\left( {n_{11} + n_{10} } \right)}} = \frac{{m_2 }}{{n_1 }}\]

Because it is possible for no marks to be recaptured (\(m_2=0\)), the same adjustments to the MLE can be made as found in the other estimators presented in Table 1.

The information matrix (the negative of the expected value of the second derivative matrix) is found by differentiating the first derivative or difference equations and replacing the observed statistics (the number of fish with each capture history) with its expected value. If the parameters are ordered by \(N\), \(p_1\), and \(p_2\), the information matrix is: \[\left[ {\begin{array}{*{20}c} {\frac{{1 - \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{N\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}} & {\frac{1}{{(1 - p_1 )}}} & {\frac{1}{{(1 - p_2 )}}} \\ {\frac{1}{{(1 - p_1 )}}} & {\frac{N}{{p_1 (1 - p_1 )}}} & 0 \\ {\frac{1}{{(1 - p_2 )}}} & 0 & {\frac{N}{{p_2 (1 - p_2 )}}} \\ \end{array}} \right]\]

The inverse of this matrix provides the variance-covariance matrix of the estimators: \[\left[ {\begin{array}{*{20}c} {\frac{{N\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{p_1 p_2 }}} & {\frac{{ - \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{p_2 }}} & {\frac{{ - \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{p_1 }}} \\ {\frac{{ - \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{p_2 }}} & {\frac{{p_1 (1 - p_1 )}}{{Np_2 }}} & {\frac{{\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{N}} \\ {\frac{{ - \left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{{p_1 }}} & {\frac{{\left( {1 - p_1 } \right)\left( {1 - p_2 } \right)}}{N}} & {\frac{{p_2 (1 - p_2 )}}{{Np_1 }}} \\ \end{array}} \right]\]

The precision of the estimated abundance is
\[se\left( {\hat N} \right) = \sqrt {N\frac{{\left( {1 - p_1 } \right)}}{{p_1 }} \frac{{\left( {1 - p_2 } \right)}}{{p_2 }}}\] If the capture probabilities are small, then \(\frac{1-p_i}{p_i}\) is large, and the precision is poor. An estimated \(SE\) is found by replacing the unknown parameters by their estimators to give: \[\widehat{se}\left( {\hat N} \right) = \sqrt {\frac{{n_1 n_2 }}{{m_2 }}\frac{{\left( {n_2 - m_2 } \right)}}{{m_2 }}\frac{{\left( {n_1 - m_2 } \right)}}{{m_2 }}}\] which again is very similar to the results presented in Table 1. Again, similar sorts of adjustments can be made (adding 1 to various terms) to avoid problems with zero counts.

18.2 Conditional multinomial model

The Huggins (1989) conditional multinomial model is most useful when the capture probabilities are modeled as functions of individual covariates and is not needed for the basic Petersen estimator. However, this development that follows will show how the basic conditional model is developed.

Now only animals that are seen at least once are used in the likelihood. The statistics for this formulation are the number with each capture history, \(n_{10}, n_{01}, n_{11}\). Assuming no losses on capture, the conditional probability of each of these histories is found by normalizing the probabilities found in the previous section, by the total probability of been seen in the study.

\[P\left( {\left\{ {10} \right\}} \right) = \frac{p_1 \left( {1 - p_2 } \right)}{1-(1-p_1)(1-p_2)}\] \[P\left( {\left\{ {01} \right\}} \right) = \frac{\left( {1 - p_1 } \right)p_2}{1-(1-p_1)(1-p_2)}\] \[P\left( {\left\{ {11} \right\}} \right) = \frac{p_1 p_2}{1-(1-p_1)(1-p_2)}\]

Notice that the probability of history (00) is not needed.

The conditional likelihood function is a multinomial distribution among these observed capture histories and probabilities: \[L= \binom{n_{obs}}{n_{01},n_{10},n_{11}} \times \left[ {\frac{{p_1 \left( {1 - p_2 } \right)}}{1-(1-p_1)(1-p_2)}} \right]^{n_{10}} \left[ {\frac{{\left( {1 - p_1 } \right)p_2 }}{1-(1-p_1)(1-p_2)}} \right]^{n_{01} } \left[ {\frac{{p_1 p_2 }} {1-(1-p_1)(1-p_2)}} \right]^{n_{11} } \]

where \(n_{obs}=n_{01} + n_{10} + n_{11}\). Now no unknown parameter appears in the combinatorial term and the population abundance is not modeled..

The estimators are found by maximizing the \(log(L)\) by taking first-derivatives, setting these to zero, and solving. The conditional maximum likelihood estimates for \(p_1\) and \(p_2\) are found to be the same as in the complete multinomial case:

• \(\widehat{p}_1 = \frac{n_{11} }{\left( {n_{11} + n_{01} } \right) }\)
• \(\widehat{p}_2 = \frac{n_{11} }{\left( {n_{11} + n_{10} } \right) }\)

The information matrix (the negative of the expected value of the second derivative matrix) is found by differentiating the first derivative or difference equations and replacing the observed statistics (the number of fish with each capture history) with its expected value. The inverse of this gives the covariance matrix of the estimates. If the parameters are ordered by \(p_1\), and \(p_2\), the covariance matrix is:

\[\begin{equation} {Var}_{cond}\left[ {\begin{array}{c} {\hat p_1^{cond} } \\ {\hat p_1^{cond} } \\ \end{array}} \right]= \left[ { \begin{array}{cc} \frac{p_1 \left( {1 - p_1 } \right) }{n_{obs}p_2/{\left( {1 - \left({1-p_1}\right)\left({1-p_2}\right)}\right)}}& \frac{\left( {1 - p_1 } \right) \left( {1 - p_2 } \right)}{n_{obs}/{\left( {1 - \left({1-p_1}\right)\left({1-p_2}\right)}\right)}}\\ \frac{\left( {1 - p_1 } \right) \left( {1 - p_2 } \right)}{n_{obs}/{\left( {1 - \left({1-p_1}\right)\left({1-p_2}\right)}\right)}} & \frac{p_2 \left( {1 - p_2 } \right) }{n_{obs}p_1/{\left( {1 - \left({1-p_1}\right)\left({1-p_2}\right)}\right)}}\\ \end{array} }\right] \end{equation}\]

The variances of the capture probabilities are very similar to those in the complete multinomial models being of the form \(\frac{p_i(1-p_i)}{\textit{Expected number of fish captured at the other occasion}}\). There is asymptotically no loss in information about the capture probabilities by using the conditional model – again this is not surprising as the unknown number of fish never seen can’t provide information about the capture probabilities.

The estimated variances are found by substituting in the estimated capture probabilities and reduce to a surprisingly simple form: \[\widehat{{\mathop{\textrm{var}}} }_{cond} \left[ {\begin{array}{*{20}c} {\hat p_1^{cond} } \\ {\hat p_1^{cond} } \\ \end{array}} \right] = \left[ {\begin{array}{*{20}c} {\frac{{n_{11} n_{01} }}{{\left( {n_{01} + n_{11} } \right)^3 }}} & {\frac{{n_{11} n_{10} n_{01} }}{{\left( {n_{01} + n_{11} } \right)^2 \left( {n_{10} + n_{11} } \right)^2 }}} \\ {\frac{{n_{11} n_{10} n_{01} }}{{\left( {n_{01} + n_{11} } \right)^2 \left( {n_{10} + n_{11} } \right)^2 }}} & {\frac{{n_{11} n_{10} }}{{\left( {n_{10} + n_{11} } \right)^3 }}} \\ \end{array}} \right]\]

Huggins (1989) showed that an estimator for the abundance can be obtained as: \[\widehat{N}_{cond} = \sum\limits_{\textrm{{obs~animals}}}^{} {\frac{1}{{\hat p}_a^*}}\] where \(\hat{p}^*_a\) is the probability of seeing animal \(a\) in the study. In this simple Petersen study all animals have the same probability of being seen in the study, namely \(1-(1-p_1)(1-p_2)\) and the estimated abundance reduces to: \[\widehat{N}_{cond} = \frac{{n_{obs} }}{{1 - \left( {1 - \hat p_1 } \right)\left( {1 - \hat p_2 } \right)}} = \frac{{\left( {n_{10} + n_{11} } \right)\left( {n_{01} + n_{11} } \right)}}{{n_{11} }} = \frac{{n_1 n_2 }}{{m_2 }}\] which is the familiar Petersen estimator.

Huggins (1989) also gave an expression for the estimated variance of \(\widehat{N}\): \[ \widehat{\textrm{var} }\left( {\widehat{N}_{cond} } \right) = \sum\limits_{{\textrm{obs animals}}}^{} {\frac{{1 - \hat p_a^* }}{{\left( {\hat p_a^* } \right)^2 }}} + \widehat{D}^T \hat I^{ - 1} \widehat{D} \] where \(\widehat{D}\) is the partial derivative of the estimator \(\widehat{N}\) with respect to the catchabilities \(p_1\) and \(p_2\), and \(\widehat{I}^{-1}\) is the estimated covariance matrix of the estimated catchabilities.

Long and tedious algebra reduces to the familiar form: \[ \widehat{\textrm{var}}\left( {\widehat{N}_{cond} } \right) = \frac{{n_{01} n_{10} \left( {n_{01} + n_{11} } \right)\left( {n_{10} + n_{11} } \right)}}{{n_{_{11} }^3 }} = \frac{{n_1 n_2 (n_1 - m_2 )(n_2 - m_2 )}}{{m_2^3 }} \]

The conditional multinomial estimator is fully efficient – a conclusion echoed by Link and Barker (2005) who showed that there is no loss in information in using a conditional multinomial model to estimate abundance in the more general open-population capture-recapture setting.

For these reasons, we will ONLY use the conditional multinomial model in the Petersen package following the work of Yee et al (2015).

19 Appendix B - Using the VGAM package

Yee et al. (2015) developed the VGAM package which includes functions for fitting closed-population capture-recapture models, of which the Petersen-type studies are special cases. We will illustrate the use of this package with the Rödli Tarn data.

19.1 Data structure

The data structure to use the VGAM functions is slightly different from the data structure used by the Petersen package.

A data.frame is constructed with

• Two variables (traditionally called t1 and t2) for the capture history which must be numeric columns with 0 representing not captured and 1 representing captured. The Petersen package includes the split_cap_hist() function that can split the capture history represented by a single string.
• Each animal must represented by its own record. Hence, grouped capture histories with a freq variable must be expanded to individual records.

See details in the examples below.

Yee et al. (2015) provides supplemental materials with code examples from this the following were derived.

19.2 Example Rodli Tarn

This was analyzed using the Petersen package in Section 3.5.1.

In this example, we first demonstrate how to split the capture histories and expand capture histories to individual records required for the VGAM functions.

library(VGAM)

data(data_rodli)

# split the capture histories
rodli.vgam <- cbind(data_rodli, 
                    Petersen::split_cap_hist(data_rodli$cap_hist, 
                    make.numeric=TRUE))

# need to expand capture history by frequency
rodli.vgam.expand <- plyr::adply(rodli.vgam, 1, function(x){
  x[ rep(1, x$freq),]
})

head(rodli.vgam.expand)
##   cap_hist freq t1 t2
## 1       11   57  1  1
## 2       11   57  1  1
## 3       11   57  1  1
## 4       11   57  1  1
## 5       11   57  1  1
## 6       11   57  1  1

Then the traditional Petersen estimator is the Mt model (in the parlance of the closed-population capture-recapture models).

rodli.vgam.m.t <- vglm(cbind(t1,t2) ~ 1,
            posbernoulli.t, data = rodli.vgam.expand)
summary(rodli.vgam.m.t)
## 
## Call:
## vglm(formula = cbind(t1, t2) ~ 1, family = posbernoulli.t, data = rodli.vgam.expand)
## 
## Coefficients: 
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept):1 -0.74444    0.16086  -4.628  3.7e-06 ***
## (Intercept):2  0.09181    0.19177   0.479    0.632    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Names of linear predictors: logitlink(E[t1]), logitlink(E[t2])
## 
## Log-likelihood: -233.9047 on 456 degrees of freedom
## 
## Number of Fisher scoring iterations: 4 
## 
## No Hauck-Donner effect found in any of the estimates
## 
## 
## Estimate of N:  338.474 
## 
## Std. Error of N:  25.496 
## 
## Approximate 95 percent confidence interval for N:
## 288.5 388.45

19.3 Example Rodli Tarn - Chapman correction

This was analyzed using the Petersen package in Section 3.5.2.

library(VGAM)

data(data_rodli)

# add extra animal that is tagged and recaptured
rodli.chapman <- plyr::rbind.fill(data_rodli,
                    data.frame(cap_hist="11", freq=1, comment="Added for Chapman"))

# split the capture history
rodli.vgam.chapman <- cbind(rodli.chapman, 
                            Petersen::split_cap_hist(rodli.chapman$cap_hist, 
                            make.numeric=TRUE))

# need to expand capture history by frequency
rodli.vgam.chapman.expand <- plyr::adply(rodli.vgam.chapman, 1, function(x){
  x[ rep(1, x$freq),]
})

head(rodli.vgam.chapman.expand)
##   cap_hist freq comment t1 t2
## 1       11   57    <NA>  1  1
## 2       11   57    <NA>  1  1
## 3       11   57    <NA>  1  1
## 4       11   57    <NA>  1  1
## 5       11   57    <NA>  1  1
## 6       11   57    <NA>  1  1

# fit the Mt model with the Chapman correction
rodli.vgam.M.t.chapman <- vglm(cbind(t1,t2) ~ 1,
            posbernoulli.t, data = rodli.vgam.chapman.expand)
summary(rodli.vgam.M.t.chapman)
## 
## Call:
## vglm(formula = cbind(t1, t2) ~ 1, family = posbernoulli.t, data = rodli.vgam.chapman.expand)
## 
## Coefficients: 
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept):1  -0.7270     0.1599  -4.546 5.46e-06 ***
## (Intercept):2   0.1092     0.1910   0.572    0.567    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Names of linear predictors: logitlink(E[t1]), logitlink(E[t2])
## 
## Log-likelihood: -235.2889 on 458 degrees of freedom
## 
## Number of Fisher scoring iterations: 4 
## 
## No Hauck-Donner effect found in any of the estimates
## 
## 
## Estimate of N:  337.586 
## 
## Std. Error of N:  25.024 
## 
## Approximate 95 percent confidence interval for N:
## 288.54 386.63

19.4 Example - Rodli Tarn - Equal capture probabilities

This was analyzed by the Petersen package in Section 3.5.3.

library(VGAM)

data(data_rodli)

# split the capture history
rodli.vgam <- cbind(data_rodli, 
                    Petersen::split_cap_hist(data_rodli$cap_hist, 
                    make.numeric=TRUE))

# need to expand capture history by frequency
rodli.vgam.expand <- plyr::adply(rodli.vgam, 1, function(x){
  x[ rep(1, x$freq),]
})

# fit the model
rodli.vgam.m.0 <- vglm(cbind(t1,t2) ~ 1,
            posbernoulli.t(parallel = TRUE ~ 1), data = rodli.vgam.expand)
summary(rodli.vgam.m.0)
## 
## Call:
## vglm(formula = cbind(t1, t2) ~ 1, family = posbernoulli.t(parallel = TRUE ~ 
##     1), data = rodli.vgam.expand)
## 
## Coefficients: 
##             Estimate Std. Error z value Pr(>|z|)   
## (Intercept)  -0.4113     0.1528  -2.691  0.00712 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Names of linear predictors: logitlink(E[t1]), logitlink(E[t2])
## 
## Log-likelihood: -247.7207 on 457 degrees of freedom
## 
## Number of Fisher scoring iterations: 4 
## 
## No Hauck-Donner effect found in any of the estimates
## 
## 
## Estimate of N:  358.754 
## 
## Std. Error of N:  28.577 
## 
## Approximate 95 percent confidence interval for N:
## 302.74 414.76

# aic table 
library(AICcmodavg)

AICcmodavg::aictab(list(rodli.vgam.m.0, 
                        rodli.vgam.m.t))
## 
## Model selection based on AICc:
## 
##      K   AICc Delta_AICc AICcWt Cum.Wt      LL
## Mod2 2 471.86        0.0      1      1 -233.90
## Mod1 1 497.46       25.6      0      1 -247.72

19.5 Example - Northern Pike - Stratification by Sex

This was analyzed by the Petersen package in Section 6.2.

library(VGAM)

data(data_NorthernPike)
# split the capture history
NorthernPike.vgam <- cbind(data_NorthernPike, 
                           Petersen::split_cap_hist(data_NorthernPike$cap_hist, 
                           make.numeric=TRUE))

nop.vgam.time       <- vglm(cbind(t1,t2) ~ 1,   
                            posbernoulli.t,
                            data = NorthernPike.vgam)
nop.vgam.sex.time   <- vglm(cbind(t1,t2) ~ Sex, 
                            posbernoulli.t, 
                            data = NorthernPike.vgam)
nop.vgam.sex.p.time <- vglm(cbind(t1,t2) ~ Sex, 
                            posbernoulli.t(parallel = TRUE ~ Sex), 
                            data = NorthernPike.vgam)

# aic table 
library(AICcmodavg)

AICcmodavg::aictab(list(nop.vgam.time, 
                        nop.vgam.sex.time, 
                        nop.vgam.sex.p.time))
## 
## Model selection based on AICc:
## 
##      K     AICc Delta_AICc AICcWt Cum.Wt       LL
## Mod1 2  7408.68       0.00   0.71   0.71 -3702.34
## Mod2 3  7410.46       1.78   0.29   1.00 -3702.23
## Mod3 2 12143.55    4734.86   0.00   1.00 -6069.77

It is not possible to fit a model where the capture probabilities are equal at the first or second sampling event using the VGAM package.

19.6 Example - Northern Pike - Stratification by Length Class

library(VGAM)

data(data_NorthernPike)
# split the capture history
NorthernPike.vgam <- cbind(data_NorthernPike, 
                           Petersen::split_cap_hist(data_NorthernPike$cap_hist, 
                          make.numeric=TRUE))

# create the length classes
NorthernPike.vgam$length.class <- car::recode(NorthernPike.vgam$length,
                                  " lo:20='00-20';
                                    20:25='20-25';
                                    25:30='25-30';
                                    30:35='30-35';
                                    35:hi='35+'  ")

nop.vgam.time                <- vglm(cbind(t1,t2) ~ 1,   
                                     posbernoulli.t, 
                                     data = NorthernPike.vgam)
nop.vgam.length.class.time   <- vglm(cbind(t1,t2) ~ length.class, 
                                     posbernoulli.t, 
                                     data = NorthernPike.vgam)
nop.vgam.length.class.p.time <- vglm(cbind(t1,t2) ~ length.class, 
                                     posbernoulli.t(parallel = TRUE ~ length.class), 
                                     data = NorthernPike.vgam)

# aic table 
library(AICcmodavg)

AICcmodavg::aictab(list(nop.vgam.time, 
                        nop.vgam.length.class.time, 
                        nop.vgam.length.class.p.time))
## 
## Model selection based on AICc:
## 
##      K     AICc Delta_AICc AICcWt Cum.Wt       LL
## Mod2 6  7368.04       0.00      1      1 -3678.01
## Mod1 2  7408.68      40.64      0      1 -3702.34
## Mod3 5 12101.12    4733.09      0      1 -6045.56

It is not possible to fit a model where the capture probabilities are equal at the first or second sampling event using the VGAM package.

20 Appendix C. Using RMark/MARK

MARK (White and Burnham, 1999) is a Windows-based program for the analysis of capture-recapture experiments that can analyze many different types of studies RMark (Laake, 2013) is an R front-end that allows for a formula-type specification for models (rather than the graphical interface used in MARK), but then calls MARK for actual computation.s

In this section, we will demonstrate how to use RMark/MARK for the analysis of basic capture-recapture experiments.

20.1 Data structure

The data structure to use the RMark/MARK functions is similar to the data structure used by the Petersen package.

A data.frame is constructed with

• variable ch representing the capture history (e.g., 01, 10, 11)
• a variable freq representing the number of animals with that history
• other variables indicating groups and covariates.

See details in the examples below.

20.2 Example Rodli Tarn

This was analyzed using the Petersen package in Section 3.5.1.

We first get the Rodli data and then change the capture history vector variable name:

library(RMark)

data(data_rodli)
data_rodli.mark <- plyr::rename(data_rodli, c("cap_hist"="ch"))
data_rodli.mark
##   ch freq
## 1 11   57
## 2 10   52
## 3 01  120

We will used the Huggins-conditional closed-population likelihood models.

# We will be fitting some closed population models the "Closed" models, so we need to know
# the parameters of the model

setup.parameters("Huggins", check=TRUE) # returns a vector of parameter names (case sensitive)
## [1] "p" "c"

# Notice that there is NO parameter for the population size in the Huggins model

rodli.rmark.mt <- mark( data_rodli.mark, invisible=TRUE, 
                        model="Huggins",
                        model.name="model ~..time",
                        model.parameters=list(p=list(formula=~time, share=TRUE))
              )
## 
## Output summary for Huggins model
## Name : model ~..time 
## 
## Npar :  2
## -2lnL:  467.8095
## AICc :  471.8358
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -0.7444405 0.1608639 -1.0597337 -0.4291472
## p:time2        0.8362480 0.1660244  0.5108402  1.1616559
## 
## 
## Real Parameter p
##          1         2
##  0.3220339 0.5229358
## 
## 
## Real Parameter c
##          2
##  0.5229358

This gives much output, but we extract the relevant parts:

summary(rodli.rmark.mt, se=TRUE)
## Output summary for Huggins model
## Name : model ~..time 
## 
## Npar :  2
## -2lnL:  467.8095
## AICc :  471.8358
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -0.7444405 0.1608639 -1.0597337 -0.4291472
## p:time2        0.8362480 0.1660244  0.5108402  1.1616559
## 
## 
## Real Parameter p
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## p g1 t1              1         1 0.3220339 0.0351211 0.2573604 0.3943300      
## p g1 t2              2         2 0.5229358 0.0478409 0.4294597 0.6148324      
## 
## 
## Real Parameter c
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## c g1 t2              3         2 0.5229358 0.0478409 0.4294597 0.6148324
rodli.rmark.mt$results$real
##          estimate        se       lcl       ucl fixed note
## p g1 t1 0.3220339 0.0351211 0.2573604 0.3943300           
## p g1 t2 0.5229358 0.0478409 0.4294597 0.6148324

As before, the estimate of the population size is obtained using a Horvitz-Thompson-like estimator and is not part of the likelihood. These are known as derived parameters in RMark/MARK:

rodli.rmark.mt$results$derived
## $`N Population Size`
##   estimate       se      lcl      ucl
## 1 338.4737 25.49646 298.7707 400.7696

20.3 Example Rodli Tarn - Chapman correction

This was analyzed using the Petersen package in Section 3.5.2.

library(RMark)

data(data_rodli)
data_rodli.mark <- plyr::rename(data_rodli, c("cap_hist"="ch"))

# add extra animal that is tagged and recaptured
data_rodli.chapman.mark <- plyr::rbind.fill(data_rodli.mark,
                    data.frame(ch="11", freq=1, comment="Added for Chapman"))

# fit the Mt model with the Chapman correction
rodli.rmark.mt.chapman <- mark( data_rodli.chapman.mark, invisible=TRUE, 
                model="Huggins",
                model.name="model ~..time",
                model.parameters=list(p=list(formula=~time, share=TRUE))
              )
## 
## Output summary for Huggins model
## Name : model ~..time 
## 
## Npar :  2
## -2lnL:  470.5777
## AICc :  474.604
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -0.7270487 0.1599210 -1.0404938 -0.4136037
## p:time2        0.8362480 0.1660244  0.5108402  1.1616559
## 
## 
## Real Parameter p
##          1         2
##  0.3258427 0.5272727
## 
## 
## Real Parameter c
##          2
##  0.5272727
summary(rodli.rmark.mt.chapman, se=TRUE)
## Output summary for Huggins model
## Name : model ~..time 
## 
## Npar :  2
## -2lnL:  470.5777
## AICc :  474.604
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -0.7270487 0.1599210 -1.0404938 -0.4136037
## p:time2        0.8362480 0.1660244  0.5108402  1.1616559
## 
## 
## Real Parameter p
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## p g1 t1              1         1 0.3258427 0.0351297 0.2610547 0.3980483      
## p g1 t2              2         2 0.5272727 0.0476022 0.4341067 0.6185773      
## 
## 
## Real Parameter c
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## c g1 t2              3         2 0.5272727 0.0476022 0.4341067 0.6185773
rodli.rmark.mt.chapman$results$real
##          estimate        se       lcl       ucl fixed note
## p g1 t1 0.3258427 0.0351297 0.2610547 0.3980483           
## p g1 t2 0.5272727 0.0476022 0.4341067 0.6185773
rodli.rmark.mt.chapman$results$derived
## $`N Population Size`
##   estimate       se      lcl      ucl
## 1 337.5862 25.02399 298.6072 398.7109

We need to remove the Chapman model because the data is different when we compare model using AIC (later)”

rm(rodli.rmark.mt.chapman)
cleanup(ask=FALSE)

20.4 Example - Rodli Tarn - Equal capture probabilities

This was analyzed by the Petersen package in Section 3.5.3.

rodli.rmark.m0 <- mark( data_rodli.mark, invisible=TRUE, 
                model="Huggins",
                model.name="model ~1",
                model.parameters=list(p=list(formula=~1, share=TRUE))
              )
## 
## Output summary for Huggins model
## Name : model ~1 
## 
## Npar :  1
## -2lnL:  495.4414
## AICc :  497.4501
## 
## Beta
##                estimate        se       lcl       ucl
## p:(Intercept) -0.411296 0.1528326 -0.710848 -0.111744
## 
## 
## Real Parameter p
##          1         2
##  0.3986014 0.3986014
## 
## 
## Real Parameter c
##          2
##  0.3986014

We now can do the usual AIC model averaging on the two models

# Collect results and get the AIC tables
rodli.results <- collect.models( type="Huggins")
rodli.results
##           model npar     AICc DeltaAICc       weight Deviance
## 2 model ~..time    2 471.8358   0.00000 9.999973e-01 2037.917
## 1      model ~1    1 497.4501  25.61429 2.741112e-06 2065.549

20.5 Example - Northern Pike - Stratification by Sex

This was analyzed by the Petersen package in Section 6.2.

We create the proper data structures:

data(data_NorthernPike)
data_nop.mark <- plyr::rename(data_NorthernPike, c("cap_hist"="ch"))
head(data_nop.mark)
##   ch length Sex freq
## 1 01  23.20   M    1
## 2 01  28.89   F    1
## 3 01  25.20   M    1
## 4 01  22.20   M    1
## 5 01  25.00   M    1
## 6 01  24.70   M    1

# We will be fitting some closed population models the "Closed" models, so we need to know
# the parameters of the model

setup.parameters("Huggins", check=TRUE) # returns a vector of parameter names (case sensitive)
## [1] "p" "c"

# Notice that there is NO parameter for the population size in the Huggins model

nop.sex.time <- mark( data_nop.mark, invisible=TRUE, 
                model="Huggins",
                groups=c("Sex"),  # note name must match case and that used in convert.inp command
                model.name="model sex*..time",
                model.parameters=list(p=list(formula=~Sex*time, share=TRUE))
              )
## 
## Output summary for Huggins model
## Name : model sex*..time 
## 
## Npar :  4
## -2lnL:  7391.653
## AICc :  7399.655
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -1.7728555 0.1146487 -1.9975670 -1.5481441
## p:SexM        -0.1366869 0.1732879 -0.4763313  0.2029574
## p:time2       -2.0214971 0.0464885 -2.1126145 -1.9303797
## p:SexM:time2   0.2462124 0.0686219  0.1117135  0.3807113
## 
## 
## Real Parameter p
##                    1         2
## Group:SexF 0.1451876 0.0220025
## Group:SexM 0.1290323 0.0244869
## 
## 
## Real Parameter c
##                    2
## Group:SexF 0.0220025
## Group:SexM 0.0244869

summary(nop.sex.time, se=TRUE)
## Output summary for Huggins model
## Name : model sex*..time 
## 
## Npar :  4
## -2lnL:  7391.653
## AICc :  7399.655
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -1.7728555 0.1146487 -1.9975670 -1.5481441
## p:SexM        -0.1366869 0.1732879 -0.4763313  0.2029574
## p:time2       -2.0214971 0.0464885 -2.1126145 -1.9303797
## p:SexM:time2   0.2462124 0.0686219  0.1117135  0.3807113
## 
## 
## Real Parameter p
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## p gF t1              1         1 0.1451876 0.0142288 0.1194586 0.1753545      
## p gF t2              2         2 0.0220025 0.0023065 0.0179080 0.0270073      
## p gM t1              3         3 0.1290323 0.0146031 0.1030094 0.1604533      
## p gM t2              4         4 0.0244869 0.0029329 0.0193509 0.0309430      
## 
## 
## Real Parameter c
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## c gF t2              5         2 0.0220025 0.0023065 0.0179080 0.0270073      
## c gM t2              6         4 0.0244869 0.0029329 0.0193509 0.0309430
nop.sex.time$results$real
##          estimate        se       lcl       ucl fixed note
## p gF t1 0.1451876 0.0142288 0.1194586 0.1753545           
## p gF t2 0.0220025 0.0023065 0.0179080 0.0270073           
## p gM t1 0.1290323 0.0146031 0.1030094 0.1604533           
## p gM t2 0.0244869 0.0029329 0.0193509 0.0309430
nop.sex.time$results$derived
## $`N Population Size`
##   estimate       se      lcl      ucl
## 1 27860.51 2700.213 23140.38 33780.31
## 2 21524.51 2406.304 17382.86 26878.68

The estimated population size is derived from a HT-type estimator for each sex, and we need to find the total, manually:

cat("\n\n*** Total population size and se from model sex*time\n")
## 
## 
## *** Total population size and se from model sex*time
temp <- nop.sex.time$results$derived
sum(nop.sex.time$results$derived$`N Population Size`[,"estimate",drop=FALSE])
## [1] 49385.02
sqrt(sum(nop.sex.time$results$derived.vcv$`N Population Size`))
## [1] 3616.834

20.6 Example - Northern Pike - Model with length as a covariate

We first standardize the length variable and create a new variable for \(length^2\).

# we standardize the length and l2 variables
data_nop.mark$L <- scale(data_nop.mark$length)
data_nop.mark$L2<- data_nop.mark$L**2
head(data_nop.mark)
##   ch length Sex freq          L        L2
## 1 01  23.20   M    1 -0.7146489 0.5107230
## 2 01  28.89   F    1  0.3710705 0.1376933
## 3 01  25.20   M    1 -0.3330252 0.1109058
## 4 01  22.20   M    1 -0.9054607 0.8198591
## 5 01  25.00   M    1 -0.3711876 0.1377802
## 6 01  24.70   M    1 -0.4284311 0.1835532

Now we fit the model:

nop.mtLL <- mark( data_nop.mark, invisible=TRUE, 
                model="Huggins",
#                groups=c("Sex"),  # note name must match case and that used in convert.inp command
                model.name="model time*(L+L2)",
                model.parameters=list(p=list(formula=~time*L+time*L2, share=TRUE))
              )
## 
## Output summary for Huggins model
## Name : model time*(L+L2) 
## 
## Npar :  6
## -2lnL:  7150.663
## AICc :  7162.668
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -1.7120281 0.1064007 -1.9205734 -1.5034829
## p:time2       -1.5005707 0.0434446 -1.5857222 -1.4154193
## p:L            0.1734418 0.1311218 -0.0835569  0.4304405
## p:L2          -0.2486361 0.1391712 -0.5214118  0.0241395
## p:time2:L      0.1027038 0.0463942  0.0117711  0.1936365
## p:time2:L2    -0.5414320 0.0453611 -0.6303398 -0.4525242
## 
## 
## Real Parameter p
##          1         2
##  0.1233986 0.0179409
## 
## 
## Real Parameter c
##          2
##  0.0179409

summary(nop.mtLL, se=TRUE)
## Output summary for Huggins model
## Name : model time*(L+L2) 
## 
## Npar :  6
## -2lnL:  7150.663
## AICc :  7162.668
## 
## Beta
##                 estimate        se        lcl        ucl
## p:(Intercept) -1.7120281 0.1064007 -1.9205734 -1.5034829
## p:time2       -1.5005707 0.0434446 -1.5857222 -1.4154193
## p:L            0.1734418 0.1311218 -0.0835569  0.4304405
## p:L2          -0.2486361 0.1391712 -0.5214118  0.0241395
## p:time2:L      0.1027038 0.0463942  0.0117711  0.1936365
## p:time2:L2    -0.5414320 0.0453611 -0.6303398 -0.4525242
## 
## 
## Real Parameter p
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## p g1 t1              1         1 0.1233986 0.0124748 0.1009541 0.1500005      
## p g1 t2              2         2 0.0179409 0.0019345 0.0145176 0.0221533      
## 
## 
## Real Parameter c
##         all.diff.index par.index  estimate        se       lcl       ucl fixed
## c g1 t2              3         2 0.0179409 0.0019345 0.0145176 0.0221533
nop.mtLL$results$real
##          estimate        se       lcl       ucl fixed note
## p g1 t1 0.1233986 0.0124748 0.1009541 0.1500005           
## p g1 t2 0.0179409 0.0019345 0.0145176 0.0221533

The usual estimates of the population size are obtained:

nop.mtLL$results$derived
## $`N Population Size`
##   estimate       se     lcl      ucl
## 1 59207.34 9364.158 43877.6 81051.75

cat("\n\n*** Total population size and se from model\n")
## 
## 
## *** Total population size and se from model
sum(nop.mtLL$results$derived$`N Population Size`[,"estimate"])
## [1] 59207.34
sqrt(sum(nop.mtLL$results$derived.vcv$`N Population Size`))
## [1] 9364.158

We can now obtain predictions of the capture probabilities at the two sampling occasions:

pred.length <- seq(from=min(data_nop.mark$L), to=max(data_nop.mark$L), length.out=40)

P.by.length.mtLL <- covariate.predictions(nop.mtLL,
        data=data.frame(L=pred.length, L2=pred.length^2), indices=c(1,2))
P.by.length.mtLL$estimates[1:5,]
##   vcv.index model.index par.index         L       L2     estimate           se
## 1         1           1         1 -2.088494 4.361808 0.0407473768 0.0261971249
## 2         2           2         2 -2.088494 4.361808 0.0007200627 0.0004620906
## 3         3           1         1 -1.951501 3.808356 0.0475436972 0.0267135981
## 4         4           2         2 -1.951501 3.808356 0.0011574729 0.0006535652
## 5         5           1         1 -1.814508 3.292439 0.0549217128 0.0267342837
##            lcl         ucl fixed
## 1 0.0112911463 0.136443938      
## 2 0.0002046225 0.002530600      
## 3 0.0154645439 0.136913096      
## 4 0.0003825261 0.003496867      
## 5 0.0207375220 0.137541422

ggplot(data=P.by.length.mtLL$estimates, aes(x=L, y=estimate, color=as.factor(par.index)))+
   geom_point()+
   geom_line()+
   xlab("Standardized Length (in)")+ylab("P(capture)")+
   scale_color_discrete(name="Event")

21 Appendix D. Conditional likelihood approach to tag-loss studies.

We follow the approach of Hyan et al (2012) by looking at the case of 2 indistinguishable tags. The cases of distinguishable or a permanent tag is similar and not presented here.

Figure 28 shows the fates of fish in the tag loss model with 2 indistinguishable tags.

Figure 28: Fates of animals in population under tag-loss model with 2 indistinguishable tags.

In this diagram, the capture probabilities are denoted by \(p_1\), and \(p_2\), the probability of a single tag being applied is \(S\), and the tag loss probabilities are denoted by \(\rho\). The codes \(A\) and \(B\) denote fish with single and double tags applied in the first sample.

The probability of the capture histories are:

• \(P(1010) = p_1 S (1-\rho) p_2\) = single tag fish recaptured with tag present = \(m_A\)
• \(P(1000) = p_1 S \rho + p_1 S (1-\rho)(1-p_2)\) - single tag fish never seen again (either tag lost or not captured at event 2)
• \(P(1111) = p_1 (1-S) (1-\rho)^2 p_2\) = double tagged fish recaptured with both tags present \(m_{BB}\)
• \(P(111X) = p_1 (1-S) 2 \rho (1-\rho) p_2\) = double tagged fish recaptured with only tag present. There are two ways in which a single tag can be lost = \(m_B\).
• \(P(1100) = p_1 (1-S) \rho^2 + p_1 (1-S) 2\rho(1-\rho)(1-p_2) + p_1 (1-S) (1-\rho)^2(1-p_2)\) = double tagged fish lost both tags or retained at least one tag and was never captured at time 2
• \(P(0010) = p_1 S (1-\rho)p_2 + p_1 (1-S) \rho^2 p_2 + (1-p_1) p_2\) = probability of an apparently untagged fish being captured at event 2 = \(m_U\).

Notice that some fish can be double counted, i.e. fish that were single tagged and never seen again also includes fish that were captured at event 2 after loosing its tag; and fish that were double tagged and never seen again, also includes fish that were captured at event 2 after loosing both tags. Consequently, a full multinomial model cannot be constructed because the probabilities add to more than 1.

However, suppose we condition on being captured at event 2. This corresponds to fish with histories 1010, 1111, 111x, and 0010. This has total probability of \(p_2\).

The conditional probabilities are now

• \(P(1010~|~\textit{captured at event 2}) = p_1 S (1-\rho)\)
• \(P(1111~|~\textit{captured at event 2}) = p_1 (1-S) (1-\rho)^2\)
• \(P(111X~|~\textit{captured at event 2}) = p_1 (1-S) 2 \rho (1-\rho)\)
• \(P(0010~|~\textit{captured at event 2}) = p_1 S (1-\rho)p_2 + p_1 (1-S) \rho^2 p_2 + (1-p_1)\)

Now a conditional maximum likelihood approach can be used to estimate \(p_1\) and \(\rho\) in a similar fashion as the conditional maximum likelihood Lincoln-Petersen estimator.

Once estimates of \(p_1\) are obtained, the estimated abundance is formed using a Horvitz-Thompson type estimator

\[\widehat{N} = \sum_{\textit{animals captured at event 1}} {\frac{1}{\widehat{p}_{1i}}}\]

Estimates of the uncertainty of the abundance estimates are formed again similar to those in the conditional likelihood Petersen estimator.

The main advantage of the conditional likelihood approach is that models where the capture/tag retention parameters vary by covariates that are unknown for fish never seen can be fit again similar to the conditional likelihood Petersen estimator.

22 Appendix E: Brief theory for BTSPAS models.

The complete theory for the BTSPAS models is presented in Bonner and Schwarz (2011). This is a synopsis.

We consider the two cases – the diagonal and non-diagonal cases.

22.1 Diagonal case

In the diagonal model, the relevant data are:

• \(n_{1i}\) - the number released in temporal stratum \(i\)
• \(m_{2i}\) - the number from those releases temporal stratum \(i\) that are captured together in a single future temporal stratum
• \(u2_i\) - the number of UNmarked fish captured in a future temporal stratum along with the recaptures from releases in temporal stratum \(i\).

The capture history data can be structured as:

We can extract the two temporal strata and make a matrix of releases and recoveries of the form:

Recovery Stratum
               Never          rs1   rs2    rs3 ...   rsk    Tagged
Newly          seen again
Untagged                    u2[1]  u2[2] u2[3]  ... u2[k]  
captured  

Marking   ms1  n1[1]-m2[1]  m2[1]     0      0  ...   0     n1[1]
Stratum   ms2  n1[1]-m2[2]    0    m2[2]     0  ...   0     n1[2]
          ms3  n1[1]-m2[3]    0       0  m2[3]  ...   0     n1[3]
          ...  
          msk  n1[1]-m2[k]    0       0      0  ... m2[k]   n1[k]

Here the tagging and recapture events have been stratified into \(k\) temporal strata. Marked fish from one stratum tend to move at similar rates and so are recaptured together with unmarked fish. Recaptures of marked fish take place along the “diagonal” as shown below for the first 10 temporal strata:

Note that while this model is referred to as the diagonal model, it is not strictly necessary that recaptures/captures take place in the same temporal stratum as releases. For example, fish could be released in julian week 21 of a calendar year, but take 2 weeks to move to the next fish wheel for recapture in julian week 23. similarly, fish released in julian week 22, are then recaptured in julian week 24, etc. In this case the two-week offset is “hidden” from the program and we “pretend” that releases and recaptures take place in the same temporal stratum.

The Bayesian model has two components. First, we condition on the number of releases and model the recaptures as Binomial distributions: \[m_{2i} \sim Binomial(n_{1i}, p_{2i})\] This is similar to a fully stratified Petersen estimator. However, we assume that the recapture probabilities come from a hierarchical model on the \(logit()\) scale: \[logit(p_{2i}) \sim Normal(\mu, \sigma)\]

The advantage of the hierarchical model is that inference for the parameters in one stratum will depend on the data from all strata. For example, if the data suggests that all of the strata have \(p\)’s close to .025, then this information is used to improve estimates for strata with poor data (or no data at all). In this way, data is shared among the strata, but the parameters are still allowed to vary among strata. The disadvantage of the hierarchical approach is that it makes no adjustment for the ordering of the data – the same amount of information is shared between strata 1 and 2 as strata 1 and 10 or strata 1 and 100. Various extensions to the BTSPAS model could be developed to account for this autocorrelation, but in practice, we have not found it necessary. This follows the framework of Mantyniemi and Romakkaniemi (2002).

Second, we model the unmarked fish captured at the “second” event also by a binomial distribution with the same (re)capture probabilities \[u_{2i} \sim Binomial(U_{2i}, p_{2i})\] where \(U_{2i}\) is the population abundance of unmarked fish passing the second trap along with fish released in temporal stratum \(i\). Again, this is similar to the fully stratified-Petersen estimator. However, it seems intuitive that the number of fish passing the recapture trap will be more similar for strata that are close together and less similar for strata that are further apart. To account for this structure, we impose smoothness on the \(U_{2i}\) using a spline that allows for a very general shape.

Bonner and Schwarz (2011) chose to model \(U_{2i}\) explicitly as a smooth curve using the Bayesian penalized spline (P-spline) method of Lang and Brezger (2004). Two factors control the smoothness of a spline: the number and locations of the knots and the variation in the coefficients of the basis-function expansion. The classical P-spline method of Eilers and Marx (1996) approaches this dichotomy by fixing a large number of knot points and then penalizing the first or second order differences of the coefficients. In the original implementation, the spline curve is fit by minimizing a target function which adds the sum-of-squared residuals and a penalty term formed as the product of a smoothing parameter and the sum of the differences of the spline coefficients. Increasing the smoothing parameter places more weight on the penalty term and results in a smoother curve. Decreasing the smoothing parameter places more weight on the sum-of-squared residuals and produces a fit that comes closer to interpolating the data.

Although the spline model may reflect the trend in the daily population size, similar to a running mean, it is unlikely that \(U_{2i}\) will exactly follow a smooth curve. If the deviations from a smooth curve are small then it seems reasonable that forcing \(U_{2i}\) to be smooth will not have a large impact on the estimation of overall abundance . However, if there are large deviations from the smooth curve then forcing the \(U_{2j}\) to be smooth may severely bias the estimate of the total population size. To allow for roughness in the \(U_{2i}\) over-and-above the spline fit, the spline model is extended with an additional “error” (extra variation) term to allow the \(U_{2i}\) to vary above and below the spline.

Bonner (2008, Section 2.2.4) conducted an extensive simulation study to compare the Bayesian P-spline method with the stratified-Petersen estimator. Generally, the Bayesian P-spline method had negligible bias and its precision was at least as good as the stratified-Petersen estimator. When “perfect” data were available (i.e., many marked fish were released and recaptured in each stratum) the results from the Bayesian P-spline and the stratified-Petersen were very similar. When few marked fish were released or recaptured in each stratum, the performance of the Bayesian P-spline model depended on the amount of variation between the capture probabilities and the pattern of abundance over time. In the worst case, with large variations between the capture probabilities and abundances that followed no regular patterns, the two models continued to perform similarly. However, when the variation between the capture probabilities were smaller and the abundances followed close to a smooth curve the Bayesian P-spline produced much more precise estimates of the total population size.

22.2 Non-diagonal case

In many cases, the released fish are not recaptured in a single future temporal stratum, and recaptures takes place for a number of future strata.

This gives rise to the matrix of releases and recoveries of the form:

                                           Recovery Stratum
Newly
Untagged               u2[1]   u2[2]   u2[3]  ...                 u2[k]   u2[k,k+1]
captured  
               tagged    rs1      rs2     rs3 ...rs4                 rsk  rs(k+1)
Marking   ms1    n1[1]  m2[1,1] m2[1,2] m2[1,3] m2[1,4]      0  ...   0      0 
Stratum   ms2    n1[2]   0      m2[2,2] m2[2,3] m2[2,4] .... 0  ...   0      0 
          ms3    n1[3]   0       0      m2[3,3] m2[3,4] ...  0  ...   0      0  
          ...  
          msk    n1[k]   0       0      0  ...  0            0    m2[k,k] m2[k,k+1]  

In the above representation, released fish take between 0 and 3 weeks to travel between the two traps between release and recapture.

BTSPAS includes a function to fit a log-normal distribution to the length of time that individual fish take to move between the temporal stratum of release and temporal stratum of recovery, but this model may not be sufficiently flexible. Alternatively, BTSPAS includes a function that uses a multinomial distribution to describe the movement of fish from the temporal release stratum to the temporal recovery stratum with the restriction that this multinomial movement model slowly changes over time.

GIVE MORE DETAILS HERE

The hierarchical model for the recapture probabilities and the smoothing spline are similar in this non-diagonal non-parametric movement model.